Do permanents reenter the battlefield tapped with Brago, King Eternal?
Permanents changing zones have no memory of their previous state (meaning tapped or untapped):


  400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence. [...]


Due to Brago's ability not invoking any additional state changes to the blinked cards, they will enter the battlefield as they normally would. This means that a Sol Ring will enter the battlefield untapped, while a Worn Powerstone would enter the battlefield tapped.

Note that other cards, such as Eldrazi Displacer, do set the state of the card re-entering the battlefield, so those cards will always enter the battlefield tapped.