We need to resolve the problem by Monday. We need to keep trying to resolve the problem until Monday. Until implies that something that is happening will end on precisely on Monday. By implies that something that hasn't happened needs to happen anytime before Monday. Another example: He will be finished with his project by next week. He will be busy working on his project until next week.

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.