Learn Portuguese (Portugal): O navio sai agora, apesar de o porto estar fechado. - The ship leaves now, despite the port being closed.

In Portuguese, you cannot contract a preposition (like de) with an article (like o) if that article belongs to the subject of the following verb. Because o porto is the subject of the verb estar, the de and the o must stay separate.

The verb estar here doesn't have a special ending. Is it just the regular (impersonal) infinitive?

It is actually the Personal Infinitive. Because the subject is o porto (a third-person singular subject), the personal infinitive form is estar, which happens to look identical to the regular infinitive. If the subject were plural, like os portos, you would clearly see the personal infinitive ending: apesar de os portos estarem fechados.

Could I use the subjunctive here and say apesar de o porto esteja fechado?

No, because apesar de ends in a preposition (de). In Portuguese, prepositions are followed by the infinitive, not the subjunctive. If you wanted to express 'even though' with the subjunctive, you would need to use a conjunction like embora (embora o porto esteja fechado).

Why use estar fechado instead of ser fechado?