I don't think it would be stereotypical at all for the protagonist to be promoted to captain. Star Trek has never had a protagonist before, and they don't have a lot of people get promoted to captain. And as far as I know, it's not a common trope in other fiction.
FYI:
They usually are the Captain. The protagonist is the point of view character or characters. Sisko was promoted to Captain. You can have more than one protagonist, and or antagonist.
The three classic forms of drama: Man vs man, man vs himself, man vs nature. The protagonist is on the left side of the equation, and the antagonist on the right. Knowing which you are dealing with is a a great help in understanding what you are trying to write.