The Antico District is basically the whole bottom area of Rome. You must renovate five buildings within this area. To make sure you're in the Antico District, just go to the map and move the cursor over to where you are. If it says you're in the Antico District in the top left, you're good to go. Most buildings are around 800f to renovate, so it won't cost you all that much. Plus, if you complete La Volpe's memory in DNA Sequence 3, the building you renovate for him at the end will count towards the achievement. I got the achievement when I renovated it.
It's also good to destroy any Borgia Towers in the district since you're not allowed to renovate buildings in areas still influenced by the Towers.