* Wuher: This Ithorian was the owner and operator of the Mos Eisley Cantina in Star Wars: Episode IV - A New Hope. He wasn't strictly a bartender, but he was the one who ran the establishment and oversaw the serving of drinks.
* Dexter Jettster: This human was a bartender at the Chug's Bar on Coruscant in Star Wars: Episode II - Attack of the Clones. He was known for his eccentric personality and his knowledge of current events.
* Lagarto: This Rodian was a bartender at the Outer Rim Cantina in Star Wars: Episode II - Attack of the Clones. He was known for his grumpy demeanor and his tendency to overcharge customers.
So, while there wasn't a single, named bartender in the original trilogy, there are several characters in the expanded universe who fit the bill.