What a strange construction is it? Why does it sound so strange to me, Ukrainian fella? For instance, why "him" instead of "he". And, first of all, what does it mean? I even cannot translate it correctly.
Because that is the idiom: like takes a complement in the objective case:
Just here means "exactly". So just like him means "exactly like him".
When we say that a behavior or opinion or attitude is "just like" somebody, we are comparing that behavior/opinion/attitude to those which they typically exhibit: it is "just like" what we expect of them.
The structure here is what grammarians call a 'cleft construction', in which a phrase which occurs late in the 'basic sentence' is moved to become the complement of an It is clause, and is followed by the rest of the sentence.
So the sentence means "Being late is his typical behavior".
But it is not a simple statement of fact: it implies considerable irritation with that behavior, too. On the other hand, as J.R. points out, using the phrase with a positive behavior implies considerable admiration: "It's just like her to spend her weekends working for Habitats for Humanity."
It's him instead of he because the pronoun is an object, not the subject.
As for the meaning, let me try to give a couple rough paraphrases:
Essentially, the speaker is saying that lateness is one of his character traits – this fellow is habitually late.
I can see how it would be an odd construct the first time you run across it. It's just like English to be so confounding.
Simply - "It is just like him to be late" displays a moment of recognition. The person "him" is typically late & this has just been re-affirmed. The word "him" rather than "he" is more specific to (or towards) that person. "him" points the finger, whereas "he" describes "him" - does, can, will, is, should . . etc. The sentence(s) could say "It is just like him to be late. He is always late."