When I was getting ready for my freshman year of college, I took push ups way too seriously. I was doing 4 sets of 50 in the middle of a workout that included all kinds of body weight exercises including side to side push ups and diamond push ups and a few other push up type exercises. It didn't take long before 50 was a breeze. I think if my goal was to do 100 straight, I could have done it back then with out any other specific training. I was a middle pack runner in college, but when it came time to do core work, I was #1, which meant didily squat.
If you are a serious runner, being able to do 100 push ups is worthless. Yes, do some push ups, like 3 sets of 25-35, but any more than that isn't going to contribute to your long distance running ability. If you care more about being good at push ups, go for it. You'll probably be able to do it.