To bring someone back for land of the living, one should be in the Davy Jones's locker. Jack was there, that's why his crew traveled to end of the world to bring him back.
We can't say for sure that Barbossa ended up in the same place as Jack did. Or since all the curses/bindings etc are broken since the trident, we can't say whether there's Davy Jones's locker, and/or whether can people go there. Usually there needs to be someone (currently Will Turner), that decides one would go there or not. He is the current captain on the Flying Dutchman, who also had the control over the locker.
So we cannot say for sure that Barbossa ended up in the same place as Jack did, and so we can't say for sure that just by traveling end of the world do the trick, there could be more preparation other than we've seen in the movie.
Since he didn’t have time to promote someone to Captain he should be in that "undead" place...
Pirates doesn't work that way. Sailors can obtain ranks out of the usual rule-book. (they're pirates). But, Jack still is a captain. Either him or someone else would rise up to the rank of captain in Barbossa's ship. There's also their own guideline too(Pirate Code). so either following that or by mutiny,or by getting other sailors support or any other way, someone can become the captain. Barbossa doesn't need to promote someone, although he can. but we can't be sure that other sailors will follow him/her.