Learn Russian: У студента нет паспорта.

Because of the word нет (no / there is not). When you say someone doesn't have something using нет, the thing they don't have must be in the Genitive case. Since паспорт is a masculine noun ending in a consonant, we add -а to make it паспорта.

Why does студент also end in '-а'? Is it for the same reason?

It ends in -а because it is also in the Genitive case, but for a completely different reason! The preposition У (meaning 'at' or 'by') always requires the Genitive case. So студент becomes студента. Both nouns happen to be in the Genitive case here, but they are triggered by different words.

Where is the verb 'to have' in this Russian sentence?

Russian doesn't usually use a verb like 'to have' for possession. Instead, it uses a formula that translates literally to 'At [someone] there is no [something].' So the exact literal translation of this sentence is 'At the student there is no passport.'

If the student did have a passport, how would this sentence change?

You would use the word есть (is / exists) instead of нет, and the passport would go back to its basic dictionary form (the Nominative case) because it is no longer being negated. The affirmative sentence would be: У студента есть паспорт.