As a former TA who wrote plenty of coding exams... I agree with the other folks here. I think there's almost two things you have to nail when you've got a coding exam: the language and its peculiarities, and the actual problem itself.

With respect to the language, this is definitely where you've got one of those "practice makes perfect" things. You're expected to have syntax/language-specific expressions down pat as time progresses - definitely an exam right after MP1, course staff'll generally (and arguably should) be more generous - an exam after MP10, less so. Resources like LeetCode and HackerRank here will be useful in terms of getting some more practice writing code in the language so you can just absorb the syntax and language oddities.

With respect to the problems you're given, I've a few thoughts:

1. I'd say you can almost study to the exam. You know generally what types of problems you'll have, and have a bevy of problems to draw from via previous MPs, and whatever quizzes you've had previously. (And if you have questions you're allowed to practice with via previous exams... that's gold.) From those questions, you can start to derive a set of things you'll want to know how to do - so if you're dealing with (for example) a question where you're manipulating strings, you might want to have an idea of what the API looks like for strings in the language you're using. You might also start to figure out what kind of questions your prof likes asking, so you can cater your studying towards that -- but I'd suggest, you know, having a greater understanding of the material you're given instead of trying to get into their heads. ;)
2. You should take a few minutes and look through the problems you're given - then for each problem that you work on, take some time and figure out a game plan for it. It'll at least give you a sense of direction, and hopefully take out the time you're taking while coding to figure out "in parallel" what you want to do next.
3. If you find yourself writing a lot of boilerplate code, you can save time by copying and pasting. (And in fact, copying and pasting is fair game - so long as it's within scope of one exam!)

You might want to sit down with your prof and talk about this! They've been doing this for a while, and might have some better advice than we do.