Learn German: War der Kapitän gestern auf dem Boot? - Was the captain on the boat yesterday?

The preposition auf is a two-way preposition. It takes the dative case when it describes a fixed location (answering where?), and the accusative case for movement to a destination (answering where to?). Since the captain is already situated on the boat, we use the dative form dem Boot.

Why does gestern come before auf dem Boot?

German word order generally follows a rule of Time before Place. Expressions of time, like gestern (yesterday), usually come before expressions of place, like auf dem Boot (on the boat).

Could I use the spoken past tense (Perfekt) instead of war here?

You technically could say Ist der Kapitän gestern auf dem Boot gewesen?, but German speakers almost always use the simple past tense (war) for the verb sein (to be), even in casual spoken conversations.