When I was twenty-five years old , I saw him.

ItHe was pretty small, cute and helpless.

I'm going to break him up on Monday.

(?) Unsure what this means.

He’d have to go to university, I have to wave him's hand.

Or, “He has” if present tense.

ItHe was pretty small, cute and helpless.

Whenever you’re talking about a person, you never refer to them with the it pronoun, as it’s used for objects and animals. You should refer to a person with he/her/they depending on their gender/preferred pronouns

I'm going to breakpart with him up on Monday.

I think you’re trying to say that your son is leaving for school, and you tried to say “break up with the him”.
I’m English, when you say you break up with someone, that means you’re ending a romantic relationship. You should say something like “I’m going to part with him” instead

He haves to go university, I have to wave him's hand goodbye.

1. He/she/it pronoun = add an S to the end of the verb. This rule works with nearly every verb, but with the verb “have”, we say “has” instead
2. The correct phrase is “to wave someone goodbye”