How long will the severe pain from a fractured humerus last?

Varies widely. Sorry to hear about your fracture! if treated non-operatively, as many humeral shaft fractures are, most folks tend to feel significantly better within a couple weeks. Some don't feel better ad quickly and some folks do not go on to heal but develop a "non-union" which can be quite symptomatic. If you have this fixed surgically, the pain may decrease more quickly.