"He ended up pressing a wrong button instead of the one he intended to press."
What's the rule on how much of the redundancies I can omit at the end of the sentence?
Can I omit both "to" and "press"?
Yes- you may omit those two words, but you should go further and just stop at He ended up pressing a wrong button. Or, more typically, the wrong button. The sentence states that he pressed a button, and we know also that it was wrong, so it's not necessary to add that he did not intend to press that one.