I kind of did this in November. It might sound nutty, but trust me on this one.

You can trick your body into being able to withstand more lactic acid, and thus do more reps, by temporarily raising the blood pH. Do this breathing exercise with this guy and do push ups when he does them.

The first time I did this I hadn't worked out in a long time and if I did 15 reps I was satisfied. I got scared and stopped when I had suddenly done 22 push ups. I thought I was going to blackout or something, but it's perfectly safe. The day after I did it again and banged out 27 reps easily. I kept doing push ups daily the first thing I did when I woke up for a month and slowly improved over time. I can do 58 reps now, nothing earth shattering, but way more than I've ever been able to do.

I've noticed that I gain more strength when I do push ups daily for a couple of days and then take a brake before I do it again. In your case I would go for max reps and only do one set 2-3 times per day. I would not do several sets per day every day to reach that goal, I've tried that before and I've gained more strength by doing what I just described.

So do the breathing exercise, hold your breath, bang out as many push ups as possible while you're holding your breath and then start breathing and keep doing push ups if you can. Now chill the fuck out for a few minutes, because this is pretty brutal, and go on doing whatever you normally do. And later in the day do it again. Just one set to the max. Do this for 3 days, take one day off and do it again. Repeat this until it's time to do the real test, but make sure that you rest the day before.

Eat like a mad man, and then eat some more. Buy beta alanine, it's great for buffering lactic acid build up, take it every day. This is a big jump in strength if you're starting fresh, it's not impossible, but can be pretty hard to achieve. Good luck dude!