Learn Portuguese (Portugal): A professora vai acabar o projeto depois. - The teacher is going to finish the project later.

This sentence uses the near future, which is built just like in English: the verb ir (to go) plus an unconjugated verb (the infinitive). Vai means "she is going" and acabar means "to finish". Put them together, and you get "she is going to finish".

Could I use terminar instead of acabar here?

Absolutely. Both acabar and terminar mean "to finish". Acabar is extremely common in spoken European Portuguese, but terminar works perfectly fine too and means the exact same thing in this context.

Why is it A professora instead of O professor?

In Portuguese, professions usually change depending on the person's gender. Because the teacher is female, we use the feminine word for "the" (A) and add an -a to the end of the noun to make it professora. A male teacher would be O professor.

What is the exact meaning of depois at the end of the sentence?

Depois translates to "later" or "afterwards". When used on its own like this to talk about time, it simply tells us that the action will happen at a later point.