Yes. Those teams play in what they're calling the "First Four"...which used to be called the "play in games"...4 Lowest conference winning teams vs 4 lowest at large teams. They (when I was school) found out they got the at large bids during Selection Sunday and you're playing in the game 2 days later and then if you win you go to the next site for "The Big Dance".
So in the past, if you're an at large you found out on Selection Sunday....if not you waited around to see if you got to go to the Not in Tournament Tournament.
And I would also assume based on the fact that 6 cheerleaders/6 dancers from my University went the year they had the play in game, that the NCAA funded the trip.... I don't see why they wouldn't fund it regardless....you're still in the tournament....technically