I've decided not to do Jeju, because I haven't planned ahead enough and flights are all booked up. Plus I don't want to get stuck there on Sunday.
But if you should decide (because you can get a ferry to Jeju from Busan, and maybe a flight too but book in advance).... you can rent a mountain bike (which is what I've always used) in Jeju city. It's easier that way. PLus if you can't make it all the way around I think they'll come and pick you up. And it's all road biking, alot of it coastal. If you decide to go, here's a tip. 15 km coastal road, which is flat, is a hell of a lot shorter than 5km in land and up a hill. Don't take the short cuts!! All I brought was absolute minimal clothes. I didn't even wear a helmet..bad idea though. I bailed into a tree. Missed my head but I think I cracked my ribs because I felt pain for about 5 months or so.