Tilt4 - bodyweight training alone will get you a long way. Do varied circuit training routines several times a week, and you will get in shape. Are you still at Uni? If so, then join one or two of the martial arts classes. If US Uni's are like UK ones, there should be plenty of choice.
As for he initial question, I think that press ups are probably adequate in most cases for a martial artist. Lots of functional strength etc. One problem with hitting the weights, especially bench presses, is that you may start to expand your ribcage, and that makes weaker ribs. Press ups, keeping your elbows close to body, is better in this respect.
To break the questions down:
1. are push-ups effective?
yes
2. or is there something else that can give you better results for upper body strength?
yes, complimenting your BWE routine with some compound weight training.
Bodybuilders and athletes both lift weights, but take different approaches. Athletes go mostly for compound exercises, whereas bodybuilders do a lot of isolation. Leave those isolation ones out, and you should compliment your training without weakening.
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