"He puts his coat on."
Translation:Cuireann sé a chóta air.
The general rule is that indirect objects come after direct objects, and prepositional pronouns are typically indirect objects. One exception to this rule is when a prepositional pronoun is formed from a preposition of a phrasal verb; in this sentence, cuir ar is a phrasal verb (as English “put on” is), so your suggestion of cuireann sé air a chóta is correct, and the translation above is incorrect. Please report it as an error when opportunity allows for you.
Irish requires the preposition pronoun, not just the preposition in this case.
Cuirim mo chóta orm - "I put my coat on"
Cuireann tú da chóta ort - "You put your coat on"
Cuireann sé a chóta air - "He puts his coat on"
Cuireann sí a cóta uirthi - "She puts her coat on"