I was in the same boat as you were!! When i went for my police physical test, I had to do 20 pushups in a minute and i couldn't do one!! Not a single half way one!! So the instructor told me to work on my triceps and i started doing that from then on (been about 2 months now) and I can def. tell a huge difference. I do my triceps and benchpresses also and that works your triceps really well.
As far as your other problem, I really don't know what to say! But you will def. have time to work on your triceps and pushups by Sept., so don't worry about it.