Learn Farsi: پسره برادر منه.

No, they do three completely different things! In پسره (pesare), the -e is a spoken way to make the noun definite, like "the" or "that". In برادرِ (barādar-e), it is the ezafe linking the brother to you. In منه (mane), the -e is the spoken short form of است (ast) meaning "is".

Doesn't پسر (pesar) usually mean "boy" or "son"? Why is it translated as "guy" here?

Yes, its standard translations are "boy" or "son". However, in casual spoken Farsi, people frequently use پسر (and its definite form پسره) to mean "guy" or "dude", similar to how English speakers use it for a young man.

Where is the verb in this sentence?

The verb is attached to the very last word! In formal written Farsi, the sentence would end with the standalone word است (ast), meaning "is". In everyday spoken Farsi, است merges with the previous word and sounds like an -e suffix. So من است (man ast) becomes منه (mane).

How would this sentence look in formal written Farsi?

To write this formally, you remove the colloquial suffixes. Instead of the attached -e on پسره, you use a word like آن (ān), meaning "that", before it. And instead of the -e on , you write the full verb . It becomes: (ān pesar barādar-e man ast).