My 12 year old son has one broken finger, and his doc put him in a cast up to his elbow. is this typical treatment for just a finger?

It depends: It depends on the type of fracture and where it is. Some fractures can be treated by simple buddy taping; others require more extensive splinting and even surgery.

No: That is not typical for only a finger fracture.

Fracture: For a finger fracture usually a splint works or taping the fractured finger to adjacent finger. If one has an associated fracture of other hand bones or wrist etc may require additional cast.