Not a lotta experience in painting lines and stuff on gym floors, but have done some over the years. However, your situation is different.
I've only ever painted floors that were finished, not UNfinished.
You're going to hafta ask them to prep and sand the wood, then seal it and preferably clear coat a few coats. Then you can paint over that and then have them put a few more clear coats over top of your work. Otherwise, your paint will just soak into the grain of the wood, wick out and take many coats to become somewhat usable.