As others have said you're not the first in this situation, they've given some good tips on how to prepare, here are just a couple tips on how to keep your head high.
Having been in the Service and having two sons go through AROTC, I have seen and heard a few things.
There are those that come in and can get a decent APFT score right away, some work to improve and some figure what they can do is just good enough and they don't do anymore then that to improve, the latter always earned my respect.
The fact that you come into the program behind many of your peers physically doesn't matter, what matters is what you do once you're there. here are a few tips:
Put maximum effort during every PT
Your school will have a Gym, use it, PT will not be enough to improve your scores.
Ask for help and tips from other cadets, get a work out buddy and some running partners.
Put in the effort to improve.
What I'm getting at is this, even if you start at the bottom, if you show the desire and commitment to improve, let them see you are working harder then most to improve and never quit, you'll earn the respect of your peers. Take that extra step when you feel like you're about to fall, go for that last push up when your arms say no. Push forward with more intensity then those that already score higher. I seem to have the chant "Rudy! Rudy!" in my head now. (Old movie reference, you should see it)
Just never quit, show everyone that doing everything you can to improve, volunteer for remedial PT before the send you there. You do this, you won't have any problem keeping your head held high.
Best of luck to you.