Hi there, this is Dr. Elaine
Aspirin is really the only medication that you can give OTC. The dose (5mg/lb body weight) can be given every 12-24 hours. Use buffered aspirin, not enteric-coated (the coating can allow it to accumulate).
Warm moist compresses (like a soaking wet washcloth) may help as well.
If this seems to be worsening I would advise having him seen on emergency. Something like an infection could become quite serious. I would be especially concerned if he has or does quite eating.
Warm compresses is a great idea!! I think he may have sprung it, if it were broken, would I know? What are the signs of a broken arm?
If there were a complete fracture, the arm below the fracture would be very unstable, and he would not be bearing weight on the limb, as it would collapse under him. He most likely would be in severe acute pain any time he moved, as that is when a fracture really hurt--i.e. when the bone ends are not stabilized and impinge upon the overlying soft tissues.