That sounds complicated. Why can't he go to her office and say, "Good to run into you the other day. Want to grab lunch?"
I mean he doesn't even know if he's interested in her at this point. He only knows that she's pretty. I don't think it's weird at all to ask an acquaintance in your building if they'd like to join you for lunch. It doesn't say "I want to date you." It says "I like to eat, and I assume you like to eat too."