Heliograph for communication. Whatever the content of your Engineering pro fatui, you're still stuck with a Roman industrial base. Even the most rudimentary electronics requires at least loads of copper wire that would be prohibitively expensive and would get stolen faster than could be replaced. Mirrors of polished metal are achievable. They would even work at night, with sunlight replaced by properly managed fire, though focusing the beam would be trickier.

For transport, I suspect the Romans pushed their technological base to the limit and you couldn't do much better. But if your communication is significantly faster than that of your competitors, that gives you an advantage in transport as well, whether it be trade or war you're interested in.

• Proper horsecollars, of course.


• Invent the dandy horse. A much cheaper way to move clerks and similar personnel than keeping real horses around. (As an added benefit, bicycles and sewing machines are great motivations for your engineering industry a few decades down the line).


• Invent or improve the wheelbarrow. It is unclear if Rome had it, any anyway the chinese center-wheel design has advantages for longer stretches. That means your peasants no longer need a mule or ass to carry their loads to market.


• Take a look at food preservation, this will simplify the logistics of moving troops around the Imperium.

• Stirrup


• Stagecoach

Roman roads were indeed very good for that era and couldn't be improved. Army already had relay stations where a messenger could change horses. What was missing is an effective way of riding the horse or moving groups of people in a carriage. Stirrups were not known in antiquity, which limited the use of cavalry and made any horseback travel stressful. Fitting your army horses with stirrups would immediately improve your messengers' speed and make your cavalry much more formidable.

In antiquity, only primitive wagons and chariots were used, but nothing would prevent having a more sophisticated vehicles with suspension, culminating in an iconic stagecoach. They would not only make land travel faster and more comfortable. Staging stations can provide safe lodging for travelers, and coaches can travel with armed escort. Although this would make little difference for the military and officials (who would be armed or protected either way), it can completely revolutionize private travel. In real life, safe, fast and reliable land travel did not become reality until Renaissance era or even later.

Light wagons with suspensions

The Roman Empire's road infrastructure is pretty hard to improve, given the technological base of the time periods. The roads build by the early Empire were not improved upon until the invention of Macadam during the industrial revolution.

Roman horses were generally small and could use some improvement. Much more powerful horses were available by the end of the Middle Ages. But that improvement requires selective breeding and it is unclear exactly how much you could speed that process up.

Instead, for an immediate improvement in transport, you could develop wagon engineering. But introducing the principles of the Early Modern carriage and coach, such as light, spoked wheels and suspensions, you can reduce the weight of a wagon, thus making it more efficient to carry loads long distances on the excellent roads.

Rudders and rigging

There is no real technological barriers between the Romans and the Early Modern European ships that explored the oceans of the planet. There are two principle developments that could help here.

First is the rudder with a vertical sternpost attached with iron hinges. The second is a full-rigged ship. The specific combination of square and fore-and-aft sails was not developed until the Early Modern Age, yet the Romans had both types of sail at their disposal.

Roman grain carriers were already reaching 1000 tons or more, probably larger than any European ships until the 16th century Galleons, so they were plenty big enough to benefit from more modern steering and locomotive gear.

Transportation

The Romans were pretty good at roads (for the time) - however if you update their road building techniques to use Water-bound Macadam designs and methodology it would be a significant upgrade to the land transport - particularly for moving goods, equipment and troops in poor weather conditions. Which would give you greatly increased speed and reliability over large distances. It doesn't require much in the way of technology either - it's more in the design then anything else, and it can be done with lots of labour (something Rome had decent amounts of available given slaves and whatnot)

For non-road purposes you can easily invent the compaass a thousand or so years in advance - this will improve navigation by ship (especially when visibility is poor)

Speaking of ships - quick wins include things like the stern mounted rudder, Junk rigging, Fin keels are all things that you can introduce quickly, cheaply and with existing materials and tooling but that will give you a signficant advantage in terms of manueverability, stability, speed and military capability in the ocean. You'll establish naval superiority over your rivals without breaking a sweat.

Communication

With decent engineering knowledge it shouldn't be too difficult (over time) to essentially build up a primitve electrical telegraph system. Even the first working system as invented by Sir Francis Ronalds was capable of sending communications over distances of up to 8 miles - a relay of those set up across the territories of the empire would vastly improve communication.

While that's in progress you can use the gunpowder knowledge to set up a relay system of signal flares, it'll only give you a very limited vocabulary true - but for doing things like sending warnings about those pesky barbarians and their invasions or basic co-ordination of troop movements it could be very effective.

Communication.
Semaphores (not heliostats). Even a system that does not use lights can speed up communication from weeks to hours. One of the main destabilizing factors of the Roman Empire was military uprisings by army commanders; semaphore lines halve the reaction time, or cut it down even more since you can activate army rivals within hours.

Positional notation for numbers.

Roman numerals are horror for arithmetic. Only specialists could do it.

Our standard decimal system is one, and probably easiest to introduce. It is base-10, which makes it easy to divide by 2, and 5, and any multiplication of them; division by 5 isn't that interesting in practice, 3 is much more useful, so introduce base-6 or base-12 if the populace will accept it.

Public schooling.

Child labor is common and a necessity for many. You have half a year between "you can talk with it" and "it is needed for field work", so use that time. The amount of knowledge you can pass on is limited; take your estimation of what you can cram into that time, then reduce it because children will break down if you overdo it. Remember you can't force parents to send their children to school.

Open those schools for adults, too. Deal with those who have knowledge to strongly oppose to that. Some will do anything to block the growth of unwanted competition: PR, slander, arson, murder, judicial system abuse.

You will have to budget for this, but distributing knowledge is the fastest way to get something into motion (also the most uncontrollable one - be prepared to deal with, erm, "inappropriate" use of the knowledge, that's a pretty good plotline generator).

Police.

There was no police. Robber bands were active within towns; the wealthy needed bodyguards to get home at night, the poor weren't worth being robbed, but the middle class had to stay at home or get robbed (so the middle class had a hard time doing anything).

Communities tended to organize a militia. Know a few strong men, pay them to pay the robber of the last night a visit and take back at least the one trinket that you really cannot afford to lose. Hope the strong men aren't paid more by the robbers last night (but most robbers cannot afford to give back too much). Those strong men were essentially enforcers, who might or might not ask questions before taking out their sticks - you chose your enforcers based on what you wanted to have, and you didn't want to escalate things so far that they'd simply slay you (robbers usually don't want to slay their victims: a dead victim can't be robbed again). So it was somehow manageable, just not really safe.

Separation of Powers.

Strengthen the police and see it abused by the powerful... so you need to install a judicial system. You'll need to invest it with some religious mumbo-jumbo.
However, the Romands knew how to apply rules "sine ira et studio", i.e. neutrality and objectivity were known concepts, you'd just have to make sure that it becomes the operating principle of the judicial system (the Romans tended to stick with "what works", not so much with "the principle of the thing", so you'd need a demonstration community).

Modern (sort-of) economics.

Get somebody to work on Adam Smith's theories, and translate them to the contemporary mindset. Should work well since Romans had a pretty mercantile mindset, but they were also pretty matter-of-fact with cartels and other forms of powerplay such as murder, so if something went too far they turned to interventionism.

Not sure about human rights, or emancipation.

The "patronus" was essentially king of his family. Females were always under patronage: Of the patronus of the house they were living in (whether his family or not), or of their husband when they got married. Independent women were almost nonexistent, and had to make a living by, er, "entertaining" wealthy customers (the smarter ones went far beyond just sexual entertainment, but even these were never equals to their customers).

Either accept it as a fact of the Roman culture, or risk strong opposition (in particular, senate starting to oppose you) trying to change it.

These changes take time. Adopt a policy of travelling to the future, see what worked and what didn't, travel back and adjust. If your time-travel mumbo-jumbo allows this.