What does ‘You shouldn’t get on him about being late’ mean?

Don't scold him for arriving late.

It means do not criticize him for tardiness.

"get on someone" in this use means to berate, to verbally point out some "wrong".  The idea of the sentence is that you ought to be less severe in how you treat him about his tardiness.  Now, why they are suggesting more leniency for that person, is not clear without context.

The idea though is that you are being too harsh and what you are trying to make better, is only going to get worse by trying THAT way.  Whether this is a situation where he might retaliate and become violent, or that he might close up and simply be late more often, out of spite, or for some other reason, is open to discussion and depends very much on the situation.

It means that someone is telling you to not scold what ever person is showing up late.

🥴 They want tardiness condoned for special favors hes doing or they feel bad if hes in some kind of bind.