How are their and your bodies oriented when you're pinned? In pretty much every scenario, you should be using your hips and not upper body strength to move them/yourself. If they have you mounted, look up the "trap and roll" escape and then "shrimp" escape. If they have you pinned from the side, then you need to do a pretty violent bridge of your hips up, scoot your butt away from them, and then slide your knee between you and then. If they have you pinned from the top, i.e. North-south, well, this one sucks. They best thing you can do is stiff arm them with both arms before they get the pin. The idea here though is you're still trying to buck them up to either get the stiff arms or, if you're flexible enough, kick your legs up over you and catch them to push off of. If they have you pinned with you balled up and face down, i.e. you're in the turtle position, then you should try something called a granby roll which is again a hip based movement. As I started with, that's all the basic pinning positions with the basic escapes and none of them require upper body strength.
BTW, I highly recommend a pull up bar. They're cheap and attach to a door way. If you're stuck on building upper body strength with body weight: hand stand push ups, elevated push ups, chair push ups, plyo push-ups, diamond push ups, and dips all require nothing more than a couple of chairs or a counter.