Patong boxing gym is not really in Patong, but on the road going through the jungle between Patong and Karon. I think the road from Patong to Karon is about 5 km long, and the gym is pretty much in the middle.
So if you're not going to stay at the gym, you can stay either in Patong or Karon. You can jog to and from the gym, there are lots of steep slopes on the way so it's good exercise, but the road is narrow and heavily trafficked so watch out. If you want a motorcycle, I think most places try to rent them out for about 500 baht/day, but you can usually get them down to 300 baht/day if you're willing to haggle and agree to rent for a longer time. I'm talking about the 125 cc scooters that everybody's driving down there.
Finding a cheap room is easy if you're going there on low season, 500 baht/night for a room with air condition is pretty standard, and it's the same as with the bike, if you agree to rent it for a longer time you can haggle the price down a bit. But if you're going on peak season you might have trouble finding a room regardless of the price unless you've booked one a long time in advance.