In order to build the advanced version of the required buildings, you have to receive the plans via shipments as the game progresses.
Hopefully by now you understand how shipments work, but if not: As you progress in a match, you acquire XP that fills up a bar in the upper left hand corner (I'm playing the definitive edition layout). Whence the bar fills it will allow you click on the circle, allowing you to pick a perk from a list. That list is first determined, usually starting with "Beginner", "Land", or "Naval".
Now, in order to build all of the advanced buildings in one game, you have to manually make your own deck, which you can do either by visiting your home city from the main menu, or when you preparing the game (picking civs, map, etc.). When making the deck, select all the yellow icons that represent the required building, each saying "Advanced Mill", etc. Most are for the first age but I believe the armory is for the second.
With the deck completed, play a game, getting XP by killing enemies, getting treasure, setting up trade posts, etc. and pick all of the advanced buildings for shipments. I found some of the building automatically upgrade, like my market did, but I had to destroy and rebuild a trading post to get the advanced version.
If there is anything I need to clarify, let me know. Good luck!
Note: TheOnlyMatto says there is potentially an issue getting the achievement if using a speed cheat (as also suggested in the comments). I do not know for certain either which way but if you are having a problem getting it while using cheats, try the solution without cheats to see if that will solve your issue.