I was in a similar boat as you, but I did have a little more time to fix myself.
Keep working on push-ups, try doing a total of 100 or 200 spaced throughout the day.
Same thing for sit-ups but try shooting for 200-300.
Running sucks and I'm not very good at it, but just getting out and doing it will help you. Try running up hills and in the heat.
Being physically fit is part of army life and will be required of you as a cadet and an officer. Being in good shape for BEAST is more of a comfort than a necessity: it will make your life much easier and will boost your self confidence and leadership image. Just make it easy on yourself and find some time to workout because I sure wish I did. The bottom line is that the admissions committee considered you qualified for appointment and believe that you have what it takes to excel at USMA. Don't let anyone tell you otherwise.
As for the IOCT,
DrMom was right about the class, it's called Military Movement and that is where you will learn and practice the IOCT. Don't be afraid to ask some upperclassmen (especially your team leader) when you get done with BEAST for some guidance on the IOCT. Most cadets are willing to help out people who are struggling or who just want to know how to excel.
The best advice I can give for preparation is to work on bodyweight exercises such as Pullups and/or ankles-to-bar. If you ramp up your running, you will probably lose some weight also further helping you out.
Most importantly, enjoy your senior year and final few months of freedom. Go out with friends and create some good memories that you can carry with you through BEAST and the rest of your time at WP.