How they handled it:
Arnold tells his family that he's being picked on by a bully named "The Gooch." Mr. Drummond (his foster father) immediately signs Arnold up for a Taekwondo class and then leaves him to his own devices. At one point, Willis (Arnold's brother) convinces him that he has a "killer foot," which causes Arnold to challenge The Gooch to a fight. In the end, he finds out that he doesn't actually have a killer foot, but the bully is scared away because he believes it's true.
So keep that in mind, kids. If you have a bully up your ass, just convince him that you can kill him -- problem solved.
But in reality:
Arnold is slightly larger than a toddler. When he went down to confront the bully, had that kid not believed in magic feet, there's a pretty good chance that Arnold would be shitting out his own intestines after The Gooch beat them loose.