Below is a link to a program called 100 push-ups that focuses on building strength over the course of 7 weeks to get you to the 100 push-up requirement. Give this a try and let me know how it works for you!
One of the things that helped me was breaking it into 25 push-up chunks. After each set of 25, I get to my knees and shake the blood back into my arms and take a couple of deep breaths. This gets the oxygen into the muscles that need it. Try breathing in on your way down and out on your way up. I know it will feel counter intuitive but it really helps keep your muscles oxygenated during the push-ups.
One last tip - give your body a day to rest from push-ups (you can do forms, abs and cardio daily) after doing your 100 push-ups. Your muscles need the time to rebuild and get stronger. If you try to do the 100 every day as you train up to them, you might start going backwards in number. It happened to me and I learned that giving a day between doing my 100 was key in being able to increase consistently.
This is one of the requirements you have to start working on early but I promise you if you start now and do it consistently it is an achievable goal.
Are you feeling a bit bored doing your pushups? Here is a link to a site with 82 variations on pushups. Break the boredom by switching up how you are doing them - extra bonus, it will help develop other muscles than just your basic pushups.