So I'm feeling a bit stupid for not figuring it out or finding google searches not helpful.

The place I'm stuck at is Augusta Tower 300 AF on the 15th floor. When I get on the elevator it gives me the message: Access Key 13 required. Item has been detected in a previous time period.

What does that mean? I heard about Augusta tower 200 AF but I don't have that on my time machine thing

Augusta Tower 200 AF is apparently unlocked from Academia 400 AF somewhere (the same place 300 AF should have been unlocked). I don't know the specifics as I did it myself a long time ago, but I think I remember getting stuck in this same place. Hope that helps.