When I taught Army ROTC to college Juniors, I had to get them into shape relatively quickly for their 6 weeks of continuous summer training at an Army post. One of the mandatory university courses they took with me was phy-ed. And in addition to developing overall strength & stamina, they had to pass minimum standards for Army-sytle pushups & situps, each exercise being timed to 2 minutes.
I consulted with sports medicine specialists on campus, and here's what they advised and I did. They told me a fixed amount of daily reps was NOT a good way to train for such a specific timed requirement. And I should devise progress charts for each student, to encourage & track constant improvement.
Step 1: Perform as many pushups as possible in 2 minutes (in your case it could be untimed, but make it continuous until you must stop). Then you mark the reps on your chart (or computer or phone app).
Step 2: Multiply those reps by .8 to obtain your daily training reps. Do that many reps at least every other day at first, also nonstop in one set, that you can later make a nearly daily routine once your body adjusts, though I wouldn't make it a full 7-day week without any break.
Step 3: After 2 weeks "test" yourself by doing as many pushups as you can once again. Mark your chart.
Step 4: Multiply that new number of pushups by .8 again, which should have increased a little, to obtain your new higher daily training level.
Step 5: Continue to repeat these previous steps at 2-week intervals. Eventually you'll hit your pushup ceiling. Maintain at that 80% training level, testing and charting yourself to make sure you don't fall back.
The advantage to the 80% training load is that it's relatively easy to do, while still giving nearly the same benefits as a maximum 100% effort, and leaves you stamina for other exercise routines. You'll get good definition, tone and muscle growth, though if you really want to bulk then you're looking more at high resistance weight training. But the other nice thing about pushups is you can do them at home on days when you don't get to the gym.