Yes. There's an NPC in that place that will offer a quest (!). It's purely for entertainment and has no real requirements or challenge. Completing it will give you access to a room at the inn. While in the in, you accumulated a rested XP bonus that increases the XP earned across all classes by 50% until you use it up. This bonus accumulates even while logged out.

You can complete the quest in all 3 cities to gain access to all 3 inns. As the login message indicates, the inn in Ul'dah is currently closed because it adversely affected server stability in the overcrowded city. That may have been fixed in the maintenance today. I do not know.