I don't see anything wrong or strange with
How long before the exam did you start your preparations?
You could say
How long before the exam did you start preparing/studying for it?
Note that this is quite different from asking
How long did you study for the exam? Or, how many hours did you put in?
A person can reply to your question with this: "I started studying for the exam three weeks ago [does not mean they prepared for the exam for 3 weeks]. But after the first week, I got ill and could not study at all for the next two weeks. I just showed up to the exam today, hoping that I get lucky."
Edit: After seeing some of the comments, I think I should point out that Jason is right in that there are several different ways those sentences can be rephrased. But I don't see any one particular way that fits all three examples you give, as mentioned by Anton Sherwood.
You could say
How early do you want me to arrive [for the party]?
“How long had it been raining when the match began?” - Anton Sherwood
To me, Anton's rephrasing is perfect.
If it is a formal context, then asking
How long before the exam did you start your preparations?
can be a bit confusing because of the reason I mentioned above. Usually, we want to know how long one had studied for an exam and rarely when they started preparing.
But if you wanted to, you could ask
When did you start preparing/studying for the exam?