You're forgetting the key point that the other girl wasn't just being annoying - she was physically hurting OP. She was basically doing the same as what OP ended up doing, except for longer... and less visibly. And we haven't even begun to address the bullying aspect of it yet.

I agree with you that a punch to the face was taking things too far, but what the other girl did was also taking things too far in the first place. There's often not enough room to move the chair out of reach from someone sitting directly behind you, so that's not necessarily an option. And anyone who's ever dealt with bullying in school knows that if you take it to the teacher, a few things are going to happen: First of all, the bullying will only get worse - you 'snitched', so in the bullies' mind, you've shown you deserve it. Second, the teacher is more than likely going to either not do anything, or take the bully's side. There isn't necessarily a good way out of the situation OP was in.

All that being said - you're absolutely right that OP shouldn't have punched the other girl.