A friend was training for this. I wasn't sure how hard it would be, so decided to give it a try. My baseline was at about 40 consecutive pushups. So I kept it simple: one max set every day. I worked up to 100 in 4 weeks of doing that.
While 1 max set a day isnt a good way to train for it, it does underscore an important point: don't lose sight of the fact that it is an endurance endeavour. So:
- work in sessions where you do sets of pushups on short rest. The idea is to get yourself to do more pushups you can do consecutively in a way that isnt quite consecutive. If you're currently at 60 max, try to get 5x20 done with 15 to 20 seconds of rest. This will be impossible the first few times, but will quickly build muscular endurance. Adjust every few weeks as your max consecutive increases.
- work in some "assisted" pushups (incline pushups is one way, benchpress with less than half of your body weight is another way). The idea here is to get in multiple sets near, at, or even exceeding your max consecutive pushups.
Obviously, dont stop doing other things: once a week or so you should be doing 400 or 500 pushups broken into small sets over an hour. Other days you should be doing narrow pushups, other days wide pushups, other days do normal pushups, but scooted forward to force as much weight onto your hands as possible, which eventually you may get the strength built to do arms-only pushups.
While my "max set every day" technique worked for me to get to 100 relatively quickly, it was really just a way to get lots of beginner gains, it wasnt a good long term plan.