This was a short and easy one. I think I got lucky with a starting situation where you can conquer Greece.
That's about 90 BC I think. Basically my strategy was: spend a few turns after the start of the game building roads with the initial legions, and using my workers to chop forests to feed production to Rome quickly. I built a Granary and Barracks there and then focused on Legions and Ballistas. Founded Naples and Cartagena and built whatever I could there, they're not very useful early on.
The initial legions landed near Athens and captured it. Another stack of produced units attacked Salona from the north. After Athens, I moved to take Constantinople and Ankara. Then I spent some turns healing and used my ships to land in Egypt. They didn't have anything that could resist legions, but I made peace after taking two cities because that's all that's needed for the UHV. That turned out to be a mistake because it trapped a bunch of units in Memphis. I thought they'd collapse soon after but that turned out to be wrong.
Salona is a great production location actually, and I used it as my main military city. Rome would move to build all relevant buildings to get the most out of the UP, but never before any other city considered those buildings. Legions in the meantime. In that screenshot you can see those new units pushing into recently spawned Gaulish cities.
This is from around 200 AD, now with Egypt fully conquered, an additional city in Spain, and also one in Britain (not pictured). That only left Africa. I actually put that off because Carthage usually has like 3 war elephants and a bunch of other units there, so I wanted a proper force. Instead I was handed a convenient collapse, which made things a lot easier. Would have been able to take them anyway though.
Around that time barbarians showed up in the Balkans pretty consistently, tying up my reinforcements. I think that's pretty fun and thematic, and never too hard. One time I failed to stop them at the Alps and would up with a bunch of plundered improvements in Italy, that could have been done better. In general I was far from using my units optimally in this game.
The building goal also was a close call. Using the UP made things easier, and I also utilised Citizenship. Tech path was Currency, Law, Engineering. That meant Arenas came kind of late and I had to rely on my military production cities to quickly build them. I wonder if Engineering first would have been better considering how many Arenas are required compared to Markets, and how the latter can benefit from Citizenship. I also didn't really make a plan which cities were responsible for which buildings, and it still turned out fine.
The Empire in 320 AD, right before the Byzantine spawn. It kind of sucks that they show up to take everything right after you completely the goal, but then again it also removes a border to defend.
With the eastern half also goes 1/3 of your research, which kind of stings for the tech goal. I didn't do much to facilitate being good at research, although Republic helped: the AI was always running specialists and I ended up with one Great Scientist and two Great Merchants, who I sent to Persia for money. Which that I could run a decent number of turns on 100% and make good progress. As you can see, at that point there were only the two medieval techs missing. I went Scholarship first to have the option to build research if needed, but I don't think the order matters much. I don't think there was meaningful competition over these techs.
And the final turn of the Western Roman Empire right before completing the UHV. You can see that Gaul is threatened by a lot of barbarians at this point. I've also seen Horse Archers further east already, which suicided against my forested hill legions.