It's not automatically a full day off, he'll almost certainly get into the game later on if the situation warrants it. Since Barger was called up he's appeared in 65 of the 72 games so he is largely playing every day.
Barger has pronounced platoon splits this season as well, so it's not the end of the world if he sits for a few PA to start the game when a lefty is on the mound. He is sitting at 78 wRC+ vs LHP as opposed to 141 wRC+ vs RHP. A large part of the reason why the team has enjoyed so much sustained offensive success is due to effective platooning to maximize all of the parts of the roster.