The most commonly used expressions are: Don't get upset with me. Don't get upset at me. Don't get angry at me. Don't get angry with me. However, no preposition is wrong. English has over 100 prepositions and you can use any of them to produce a grammatically correct sentence. So your question is not the right question. Instead of asking which sentences are correct, you should instead ask which sentences are the most useful . That is a judgement call, not a grammar question. Among the responses there would be much disagreement because people don't talk alike. For example, I the sentence "Don't get upset on me" is unusual. However it is not wrong and I could even imagine saying it myself. Indeed, it is interesting to analyze its meaning. It would mean "don't get upset in my presence".

don't get upset with me