re: What’s your best conversation starter for a girl that lives in the same apartment building

Life is too short. Don't be a douchebag, but be confident and make your move.

Say hello. Introduce yourself. Make small talk. Where are you from? How long have you been here? Simple small talk. Then tell her y'all should grab a drink sometime. Get the number and hangout. If there's mutual interest she won't show you yet. You'll get it over drinks. If you have too many drinks on the first night and you take her to pound town then she's not the one for mom.