Learn Swedish: Vi måste rädda mannen innan han dör. - We must save the man before he dies.

We use innan because the main part of the sentence (Vi måste rädda mannen) is positive. If the main action was negative, such as "We can not save him before..." (Vi kan inte rädda honom förrän...), you would have to use förrän instead.

If I want to start the sentence with the time phrase innan han dör, how does the word order change?

If you put the "before" clause first, it counts as the first block of information in the sentence. The Swedish V2 rule says the main verb must come second, so the main clause has to swap its subject and verb: Innan han dör måste vi rädda mannen.

Why isn't there an att (to) before rädda?

Måste (must / have to) is a modal verb. Just like in English, Swedish modal verbs are followed directly by the infinitive form of the next verb without the word att. You say måste rädda, just like you say "must save" instead of "must to save".

The verb dör looks a bit short. How does it get that present tense ending?