During applications, how can I prove I was the first author of the papers that I couldn't become first author for various reasons?
As this is my only published paper as an undergrad, and being first author means everything for me in my application, is there anyway for me to prove to the admission committees (or the professors; whoever will review my application) that I was the main contributor of the paper?


Short answer: no. Aside from the professor writing in his letter that he did indeed not do the work and just put his name front to brush up his CV, I see no way how you can prove that you were indeed the "actual" first author. And, given that this would essentially mean that the professor confesses unethical behavior, I see very little chance of this happening.

Anyway, I am not so sure whether being first author means "everything" for you. Undergrads are often not the first authors of publications (for many reasons, not just the unethical reason that you were denied first authorship). Further, if I see a paper with a senior author and an undergrad, I am not assuming that the professor did most of the hard work and undergrad just advised - independently of how the authors are ordered. I would assume your average admission committee to be similarly realistic.