Approximately how long will a broken humerus be in a cast for?

6-8 weeks in general. Not all humerus fractures are the same and require treatments specific to each patient. Breaks in the middle of the bone are immobilized in a splint or cast or just a sling until the broken bone ends get "sticky" and begin to move slowly as one bone. Casts are converted to braces or slings once early adequate healing is confirmed.
Varies. It varies somewhat based upon a more specific description of the humerus fracture. In general, humerus fractures take about 8-10 weeks to heal. Most are treated with functional braving rather Than a cast. Don't hesitate to discuss with your doctor.
Cast? I am not aware of many circumstances that a broken humerus should be is a full cast... Various splints/braces are the preferred method of treating most non surgical humeral fractures. In general, broken bones heal in 6-8 weeks if no complications occur. If your doctor put you in a cast for a non complicated humeral fracture you may want to seek a second opinion.