Who has the bridge during duty shifts??

[Star Wolf]

And how would it work on an actual navy vessel?

On an American naval vessel, the "Officer of the Deck" is in charge of the ship, on the bridge. The Captain doesn't spend all of his work day on the bridge in any case. Neither does the XO.

This wikipedia article give the general gist of that the OOD does.

http://en.wikipedia.org/wiki/Officer_of_the_deck

During general humdrum operations, there may be no need for the Captain to set foot on the bridge for several days in fact.

The real world question then becomes on an equivalent of an Enterprise class ship who is qualified to be the Officer of The Deck. The Staff Duty Officer in an Army Brigade is normally rotated among the Lieutenants. In TOS we have seen LT Sulu in the role most often.