How long could my broken arm be in a cast?

Usually 4-6 weeks. Arm fractures usually heal in 4-6 weeks depending on the extent of fracture, type of fractuere and which arm bone or bones are affected. All things equal you should expect 4-6 weeks but your orthopedist may think differently based on follow up exam, x-rays and your individual case study. Follow up and keep open dialog with your doctor.
6-8 weeks. Most fractures of the forearm and wrist heal in about 6-8 weeks. Fractures of the upper arm and shoulder may take a bit longer, sometimes up to 10-12 weeks.