I'm sorry that you have a broken arm- how long the cast will be on, depends upon what type of a break it was, and also how long it has been on already. It's best if you ask your local hospital about this.
Usually a minimum of 2-3 weeks. However, it depends on if you follow the directions given to you by your doctor. Those who take care of an arm in a cast USUALLY have their casts removed sooner rather...
To determine this, it is necessary to know the specifics about the type of fracture, bone(s) involved, etc. The best person to ask this question to is a doctor, specifically an orthopedist, a surgeon...