I would go a different route than the above comments... working out the same muscle group every day will yield little results, as you are not giving the muscle a chance to repair and grow...
As far as push-ups go, if you can't get one, try putting your knees on the ground and go from there. Ideally you want to be able to do at least 3 sets of 10 reps. Go a couple days between workouts so your body can repair and grow.
Push-ups alone will not really be enough and your body will grow used to them over time. Try other chest exercises as well, like forward dips, bench press, cable flys and so on.
You will be getting a good workout in your martial arts class either way I'm sure though..
-Nutrition Minor in college / gym rat
Forgot to mention, holding the up position will work your core (abs), but it's not your core that lets you do push-ups, its your chest and triceps.