I don't live in an apartment building anymore, but when I did, I approached every new woman I saw. Just smile, walk right up, put out your hand and say something like, "You must be new, hi my name's Dole." or "I know I haven't seen you around here before, cuz I woulda remembered you" or "How are you? My name's Dole. Did you move in recently?" Or whatever.
Ask her what she likes to do for fun. Then while she's answering, cut in and ask her if she'd like to chill with you sometime and watch a movie in your crib. Or hers, it doesn't matter. Yours would be better though.
Don't bother asking her on a date. I never did that sh1t. They would come over, and we'd talk and watch maybe 20 minutes of the movie and we'd be fvcking.
When you talk to her for the first time, let her understand that you're interested in her for more than just hanging out. Reach out and touch her shoulder or upper arm at least once during the short conversation. I'm assuming you know how to do that stuff without her wanting to pepper spray you.