Normally I would say: I took the (or "my") exam last month. The sentence, "My exam was done last month" sounds like you finished the semester, and normally I would say, "My exams were done/finished/over last month" for such an event. If you mean you simply took an exam last month, I would stick with my first sentence, "I took the exam last month." It just sounds better. Both sentences you wrote are fine and have fairly clear meanings, but they do not sound native to me.

It doesn't sound like they are the same. In the second, passive voice sentence it isn't clear who did the exam, whereas it's clear from the first, active voice sentence that "I" did it. ;)

you should write I pass an exam so your senteces are: I passed an exam last month (it means that subject - you (who passed an exam...who did action) and object - the exam (that was passed by you)) and My exam was passed last month (i have never come across this wording))))