do well in the exam

I guess that's OK, but an American would more likely say: "Do well on the exam."

A native speaker would not construct a sentence like that, though. It would be more likely that a natives speaker would say something such as" "Good luck on your exam." or "Good luck with your exam." Writing or saying "in the exam" implies the physical location of the exam, the room in which it is taking place. "Jimmy was uncomfortable in the exam because the lecture hall was so cold."

Do well in the exam, I hope you do well in the exam, did you do well in the exam? He did well in his exam.

Using 'in' this way is absolutely standard in British English.

Not in BrE.

Thank you~