First off, 'me' is the object, not subject of a sentence. It should read 'I need to resolve the problem by Monday.' You could say 'I have until Monday to resolve the problem.' 'By' is a deadline, but you could do it any time in there. 'Until' means continuously before the end date. It's a small difference, but if they are misused it sounds funny to a native. You could say: 'The problem is so big that I will have to work all weekend until Monday, when it's due.' That still sounds a little awkward, but not much. Hopefully that helped a little. Let me know if any part of this isn't clear.