Your post is too vague. You say there's no way to negotiate around the conflict, but did you make any attempt to? How much time in advance were you aware that this conflict existed? Quite a while, I'd imagine, as you should have known your final exam times when the semester started.
So unless this is some unforeseen circumstance like a shift-change mandated by your employer at the last minute, you probably should have managed the situation in a more timely fashion.
You should've planned things out better and honestly you're going to look bad for not seeing this seasonal position through. Try to work things out with your boss, I don't understand how there would be "no way" to negotiate around finals that are (I assume) happening next month.