So it sounds like magic is a evolved biological process in this new world and your Main Characters simply doesn't have it. Infact his system doesn't interact with mana and instead uses oxygen.

Instead of being able to bind and use mana, your Character is able to leech off the mana of other peoples magic. When someone casts a strong spell near him, the mana is cast out and some of the mana binds to his blood like it would for any other person. This gives your character access to mana and magic, however he needs to slowly accumulate it from the people around him who use magic in their daily lives.

This is also how other species and people can gift him spells and magics that are one time use only. They bind the spell and required mana to his blood like their own mana. Your character can't generate more mana and can't interact with it properly, so he can only activate the mana/spell once before its gone and used up.

Now to allow him to slowly build up more power, as your Character is introduced to more magic, his body slowly changes. He is able to start harnessing the magic, as the magic interacts with and changes his body. Since he was originally a blank sheet with no mana, as he develops, he is able to combine the magics of different species and continue to use their spells. It takes him much longer to learn, because he has to figure it out from scratch and his mind is no longer developing and discovering things like a babies would be, and this is a foreign concept/feeling for him.

It may not entirely fit with a typical medieval fantasy world, but what if Bob were to receive a blood transfusion after a serious injury, and the blood he was given (being of the magical world) contained managlobins and started his body producing them itself.

The biggest issue is probably one of technology, in our world the first blood transfusion didn't happen till 1665, well over 150 years after the end of the late medieval period.
But perhaps the use of magic allows some kind of rudimentary blood transfusion to take place despite the other inferior technology level?

The upside of this explanation is that unlike absorbing mana through eating or just living in the world he would only develop magical abilities after being saved from a life threatening injury (so you can control exactly where it happens in the story).
And due to the dangerous and experimental nature of the transfusion process it's unlikely others like Bob would voluntarily undergo it in an attempt to receive magical powers unless they were really desperate.

Of course you might also have to explain how the blood types of the people in your fantasy world are compatible with Bob or other real world humans, but I think that's a fairly minor hand wave in the grand scheme of things.

'Managlobin' isn't a direct biological product of humans (or animals) like haemoglobin. It's a pervasive microbial organism with a symbiotic relationship that feeds on mana and produces magic.

Bob (or others like him) gain the ability to use magic by becoming infected with this micro-organism (perhaps from other magic users, perhaps by microbes in the wild). Some people's immune systems reject it, either becoming inert 'nulls' or dying a nasty death, whichever you prefer.

Some people keep this infection in check slightly better than others (producing natural variation in magic use), and the more magic/mana you use the more these microbes multiply explaining how people gain magical power with practice.

Bonus point: the debilitating side-effects of mana withdrawal are caused by this microbe. The headaches, pain, death are caused by chemicals released when these microbes starve, and evolved as a method for coercing their hosts into finding more mana.

It's like midichlorians...but nastier ;)

Reverse the relationship between mana and magic.

Mana is not used for magic. In fact, mana interferes with the usage of magic. The more mana you have, the harder it is to cast.

Magic users are people who are able to reduce the amount of mana in their bodies temporarily, through breathing techniques and concentration. By doing so the amount of mana in their managlobin lowers and the "magic resistance" of their bodies is also lowered.

Because people who have just cast a spell have a lower concentration of mana than they had before, it has been assumed for centuries that the body has spent mana in order to cast the spell - but this is mistaking the cause and the consequence. The correct way to describe the process is that the body could cast because it had less mana.

As for golems being animated by mana: they are not. They are animated by the life force of their creators (perhaps the golems have some of their creators' blood in them), which is something else completely different from mana. However, since that life force is alive, it has mana of its own, hence the presence of mana in golems.

As a result, Robert is not only able to use macic, he is incredibly powerful at it. The only thing preventing him from casting is lack of knowledge. Once he learns a spell, he will be able to cast it more powerfully than any mage. This can vary from comical to outright dangerous or catastrophic depending on the situation.

His approach has to be different, though. Rather than reducing the mana in his body, he has to learn how to tap real magic wlthout overdoing it. There will be no teacher available in the magic world. Robert must learn magic all by himself, through techniques unfamiliar to mages, and with a distinct learning curve. Even the nature of his spells may be different.

If you want to see a similar situation, consider the animé Naruto. In that animé's world, people use a kind of mana to cast spells. It is also tied to phisiology, so that the amount of mana everyone has is limited, replenished through breathing and eating, and complete depletion of mana causes instant death.

The protagonist, however, cannot use his own mana for spells. He taps a different, outworldy source, which is overflowing infinitely for all practical purposes. That makes spell casting much more difficult for him. Whereas other people learn to cast by making an effort to gather the right amount of power from zero for each spell, Naruto has to learn how to reduce the amount of power from infinite to the right amount. The quality of his mana source also makes most spells unavailable for him for other reasons.

Also due to power considerations, he learns his masters' signsture move, which is power ball the size of a bowling ball; when he finally peaks in his skill with that spell, his own version is large enough that he could fit inside it while standing.

Ok so you know how oxygen binds to hemoglobin and mana binds to managlobin?

Well what if I tell you (insert Morpheus) that we can make crystals of organic compounds which bind metal ions by coordination (basically share lone pairs with it's d/f orbitals) who in turn bind oxygen by co-ordinating some more (actually hemoglobin does the same with iron binding the oxygen with one iron pere heme protein and 4 heme per hemoglobin)?

You also know how Sony's Spiderman was literally bitten by a spider and somehow developed web-slinging abilities but in Marvel's Spiderman and the new PS4 Spiderman had to make their own tech and shoot special fluid out of small containers that oxidize and become sticky upon contact with air? So basically the same for mana.

Now what is required to make this depends on what is mana? We could turn to zero-point energy or lepton field density or some other pseudo scientific explanation, but let's just make our own : sub-atomic black holes.

OK I am joking with that, but let's just have mana droplets irradiated by stars that are an active biological component here. Of course, in that case, as the MC breathes this world's air, she will also start accumulating the stuff, but let's just say that his biology makes him allergic, so he needs to work with the locals to ensure he doesn't get too much of it. But of course, if he stores some for magic use outside his body, no harm done right? He won't have an allergic reaction and should someone cast a "mana vaporiser" spell to deplete people, he wouldn't be affected, just left with some useless bracelets

If he hasn't mana in his body, he uses the mana around him.

Mana should be everywhere. And Bob takes the mana around him to cast spells. The spells are not always the same, since mana around him is either stronger or weaker, or has ,if you have a system like that, different elements or properties than other mana.

Bob just has trained to or, if you're quite the funny guy, if he's drunken he can feel and use the mana around him to cast magic. There are plenty of other reasons why he could gain this ability. (You could ask another question on worldbuilding).

But: people feel, when their mana is robbed and try to keep it. So he cannot take mana from persons.

Like Naruto can collect chakra from the surrounding, Bob ... well he cannot collect but use mana instantly

I hope this helps :)

Bob is unable to use magic because he has not enough alcohol in his blood ...

Mana isn't stored in mana globin, but in alcohol. That being said all the normal people have a little amount of alcohol always in their blood, which they replenish since it's used for basic living functions. Funnyly, the bodies of the people cannot deal with an overflow of alcohol and the people easy get drunk (and maybe die earlier of alcohol).

He being a great guy can store and use much mana, but only drunk.

Also you see here thats the reason why elves can spell wonderfully good.

This was inspired by @Varad Mahashabde

One possibility is a virus.
A virus can damage DNA. So it can be used to alter the DNA.
Bob gets sick after meeting some of the indigenous people. This changes his DNA to a certain extend. Over time as his cells reproduce with the new DNA he will get mana in his body. The longer he lives the more Mana he will produce until finaly all his cells were produced with the new DNA. The more mana he has the easier it gets for him to learn and use magic.

If you talk about viruses in your story you might want to give Bob a decontamination before entering this new world. Otherwise he will introduce the viruses from earth to the indigenous people like it happened to america.

Another posibility would be a symbiant. This symbiant may like his body which has no own managlobin so there is no clash between its magic and its hosts. As the symbiant gets stronger Bob will be able to leach more mana from it.

Bob will learn to be an alchemist and a blood mage

Bob does not have managlobin. But other things around him do like plants and animals. He can use basic processes (like distillation) to harvest the managlobin from the plants in the world and, after some trial and error, he can reproduce some of the reactions that make certain kind of spells (distilled managlobin + sulfate = fireball for example). Some of the more powerful rituals will require him to acquire bodily fluids from animals and people (like blood). This could have social implications, but he might even be able to create more powerful effects if given enough basic materials.

Bob has a tapeworm

Specifically, Bob has a fantasy-world tapeworm.

Whereas, Bob cannot generate his own useable mana, the native-to-the-fantasy-world tapeworm does.

You remarked:

If he does manage to get a usable supply of mana from an outside source by some means, it would dissipate fairly quickly as it can't bind to him as is.

The seems to imply the main problem is that he isn't generating his own mana. Enter (literally) the tapeworm.

Why don't normal denizens of this world use magic-generating parasites?

Maybe they do, unintentionally. However, the reward / cost of having a little more mana and, well, a tapeworm, is going to be different for someone already generating their own mana. And maybe non-magical humans somehow stimulate parasites to produce more mana for their host.

Incidentally, whereas overusing the mana might cause withdrawal for the parasite - even lethal withdrawal - the cost would be limited to a (theoretically temporary) loss of magic for Bob.

Bob has detailled knowledge about the game he played, so he knows about items that enhance abilities either permanently or temporarily. As in any game, these items are quite rare.

Initially, since he is virtually "dead", he is not affected by spells that target living matter (like mind control or paralyze). But he is very much affected by spells that target nonliving matter (like necromancer spells). His only luck is that these spells are usually not casted in populated areas and on moving and breathing targets.

Bob needs to find a permanent magic enhancer and use it. It could be a potion he drinks that causes his body to create managlobin or a prolonged stay in a sacred cave where he inhales natural gases that enrich his body with managlobin. He should feel the effect of the enhancement like a high, feeling connected with his surroundings and experiencing everything more intense.

Bob notices that he is now affected by spells targeting living matter.

Then he needs to start training the control of mana. He should realize that the amount of managlobin in his blood is barely enough to create the smallest magical effects (no more than harmless poltergeist stuff), so he starts searching for the next magic enhancer.

During his adventure, he realizes that temporary magic enhancers work on him as well, giving him enough managlobin to cast one bigger spell for a limited amount of time. But the more managlobin in his blood, the more intense the effects of spells are on him.

That adds the strategic element to magical confrontations. Is the opponent casting a fireball and he needs to drink his mana potion now to defend against it or is the opponent casting a paralyzing spell and not drinking the mana potion protects Bob against the effect?

Perhaps the solution to your problem is magic; some powerful good witch casts a spell to transform Bob, out of kindness for something brave and selfless that Bob did. Or she gives him a device, a ring or bracelet that once he puts it on he cannot remove it, that is (magically) taking mana from the air and transfusing it into his blood.

So you have a mentor or ally or grateful stranger Bob meets; early on he has no magic, but after a week wearing this device he starts being able to do small conjurings, then more, etc. Turns out he is a natural.

Essentially since bob has no mana he is the perfect conduit which allows for a mana-like being to bind with bob. The mana-like being can then use magic through bob.

Much like a rock that becomes a golem.

Mana-like beings are something like ghosts. They are pure living mana essence.