While it's technically correct, it doesn't make any sense. You're looking for something like "How did the exam go (for you)?" or "How did you do on the exam?" or just "How was the exam?".

I think it is fine. For me, 'How did you get on in the exam?' is a very natural question to ask. This is exactly what I would say to someone who's just done an exam. It means 'How did you find it?' Was it OK? Did you manage to answer all the questions? If the results have come out, 'How did you get on in the exam?' can also mean 'Did you pass?' 'Did you get a good mark?'. 'Get along' doesn't really work, as this is used more for relationships with people. NB I wonder if this is a British English use of 'get on with'? The fact that it 'doesn't make any sense' to Sam from the US - or the people who've approved his answer - seems to suggest that the expression isn't used in this way in American English. It would be interesting to hear some other opinions.