There's an interesting thread about European medieval inns going on now.
What information do we have in the Middle East? During Muslim rules from the times of Muhammad til the end of Abbasid period (circa 7th-15th century maybe)
I know they have caravanserai along Persia and Silk Road, but do they have that too in other areas?
I'm a bit busy right now so I can't go into too much detail but coffeehouses might fit the bill for what you're looking for. Coffee and Coffeehouses: The Origins of a Social Beverage in the Medieval Near East by Ralph Hattox is an excellent and short work on the subject.