no damage -- though he will do 1d4+4 at the start of his turn automatically. He's not attacked the Bullette yet, just grabbed on to stop it moving. The fact that we achieved no damage on it in the first round is worrying, though.
You are right that the only condition is grappled. Some would argue that he could drop prone and take the bullette with him. I didn't do that because that would have given the bulette advantage on an attack on me this round, and anyone attacking at range would have disadv) but if you'd allow it (Jn drops prone as his move, takes bullette with him) he will get advantage on attacks this round. I doubt that would be enough to save him though -- I fear he will be going down his first round in combat!