Should be just fine! Like Draftrider said, these little ponies are tough. I'm 5'9, about 160, and I got on my rescue Monty (13 hand hackney - light build too) yesterday bareback. I wouldn't ride hard or all the time, but to get him broke to ride should be just fine!
Also wanted to agree, I would have never, ever guessed you were only 13. You are very well spoken and very mature for your age!