Question

1. He must have been late for the school.

2. He muse have been late for school.

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Which one is grammatical? Do we have to put 'the' before 'school'?

Answer 1

#2 is correct, except for the typo.
I'd never use "the" in this case.
However, it is correct to say

He must have been late for the bus.
A bus is a specific vehicle, but "school" is a more vague entity, implying education and other activities, and not just the building.

Similarly, you can be

late for dinner
late for work
and so on.