It will be a rough ride to the target if you don't use a lathe wih a properly aligned tailstock. You could try a drill press, drilling a hole in a block of plastic or aluminum with it securely clamped on the drill press table. This hole would be equal to the shaft diameter. Then change drills to a small center drill, and then step up in crements to your final hole size. Not perfect, but will get you close. Good luck. Isn't there a local machine shop you could appeal to?
