[susan53]

I was going to say it's fine - but I think the problem is that it doesn't mean what you want it to mean :

He should have handed over his test before all the others.
This means He was supposed to give in his test before all the other students gave in their tests (but either he didn't or the speaker isn't sure if he did or not). (Eg maybe he started at ten and they started at ten thirty, so he had to finish and hand in the test at twelve while they continued till twelve thirty. But in fact he took extra time and gave in the test at the same time as the others.) If this is what you meant, then the only mistake in your sentence is the apostrophe.

But did you actually mean that when the other students took the test, he should already have handed his test in. For example, he did the test on Monday morning, but the others did it on Monday afternoon, and he was expected to hand it in at lunchtime before they started (but again, either he didn't or the speaker isn't sure).

If this is what you mean, then your sentence doesn't work. You'd have to say : He should have handed in his test before the others started.

If, on the other hand, you wanted to say that he did hand in the test before the others, then the mistake is in "should have". The sentence should be :
He had to hand in his test before all the others. / He had to hand in his test before allthe others started.

One other point: I'd say handed in rather than handed over, but it's not wrong. It may be a UK/US difference.