So your saying that you have to use weights to build upper body strength? And pushups won't do that?
Not to the degree that weights will. It is generally assumed that you are pushing between 55 and 65% of your weight in a pushup. Assuming you are 200lbs, then you're pushing about 110 to 130lbs - Not bad, and if one has not been performing any exercise at all then they will certainly see muscular strength and endurance gains. However, getting stronger is all about progression, (certainly you can increase the reps, but there is the law of diminishing returns) and you can't exactly add weight, efficiently, to pushups in a way that will force your body to adapt that you can with iron.
Pushups will indeed help upper body strength. BUT weight lifting will signifigantly increase your gains. Especially dumbells will help isolate your triceps and chest. Don't go and grab the heaviest weight you think you can do 10 reps with tho. Your looking to increase endurance as well as strength. Do 3 sets of 20-30 on top of intermingling pushups with your runs. Heck a set of pushups in the morning is almost as good as coffee. Almost. That should help with the P/U prt.
Looks like I'm in the same boat as you guys, Damcrac and Krafty. I'm strong in the lower and upper body, with a pretty solid core that sadly is obscured by the extra pounds. Im 25 and my metabolism went from insaitiable to keep feeding me but its not going anywhere :tongue2_1. Thankfully I'm lactose intolerant so dairy is nonexistant in my diet which is pprolly why I never fully jumped off that cliff to completely outta shape. I'm just trying to lose this extra 15 and hit 195 before OCS. And I'm trying to twique my run to improve it before my prt even with the weight. And Ryan_cunningham I'm gonna try the sprints and ill letcha know how it goes.
I wouldn't necessarily recommend doing sets of 20 to 30 reps, save for maybe a drop set at the end of a workout. If you want to try it and see how it works, absolutely - Go for it. In my experience I've found that lifting heavy will provide you the muscular endurance that long sets of bodyweight exercises will, but will also allow you to handle much heavier loads. Not to mention its a much more efficient use of time.
When I was in Army ROTC and a member of Ranger Challenge, I continually competed with a buddy of mine in nearly everything. What started as 250 pushups and 250 situps a day in an attempt to score a 350 APFT on the extended scale, soon turned into 750/750 a day. It was boring, and took a little over 1.5 hours for the entire workout. When I was done, had my pushups improved? Absolutely. After a month of that, boredom took over however and I got back into the gym using weight that I can press 2-10 times. I can do more pushups now than I ever could then, and I'm much stronger.
Like I said, you're more than welcome to try whatever you want. Its how I've learned what works, for me. And perhaps that is what you will have to do because everyone responds different to specific stimuli. Find what works for you, but if you want to progress quickly, and don't have time for experimentation, lean on others who have gone before you and found what generally works the best.
And definitely let me know how those hill sprints work out for you. Tabata intervals on the erg machine might also help.