I was there nine years ago and my guide steered me away from the Mongolian eateries as not being particularly good or sanitary. In UB, there was an excellent Indian restaurant ( up on a second floor as I recall) and a Korean buffet that we went to for lunch and was quite good. Sorry, I didn't write down names. The rest of the trip I was down in the Gobi where the raw sheep's milk cheese, unpasteurized horse milk and boortsog (fried dough) made for true adventure.
As for drinks, you must have a Khan Brau. Also, bring back a few bottles (70 cl. or 50 cl.)Genkhis Khan Mongolian vodka.