Locally, you can get kinda-sorta famous-ish. As in, "Boy, that Johnnyboy pours the driest martini in Billings..." Stuff like that. Fortunately for you, this sort of homegrown fame can be infinitely more useful to you than your more "standard" definition of fame. Being well-known means more customers and bigger tips. If you're the guy everyone wants to order their drink from, expect to see your annual salary go up accordingly.
But if it's paparazzi and limo comps you're after, you're barking up the wrong tree. Unless you become a movie star or discover a cure for cancer in your spare time away from bartending, it's likely that fame won't be part of your life.