It's been nearly a decade since I've been to Mongolia, so take all this with a grain of salt.

Six days is enough time for a couple of excursions outside of UB. Kara Korum / Erdene Zhu (the ruins of the city associated with Chingis Khan) is worth a visit and you can stop at a National Park on the way where you can see the teki (Przwalski wild horse). Usually this gets done as a 2 day / one night trip, with the night spent at a ger camp.

With a Mongolian-speaking interpreter, you can drive up to pretty much any ger (the preferred word in Mongolia; yurt is a Tuvan word) and be invited in for a visit. Typically you have to eat dry cheese, drink araq (fermented mare's milk), and sniff your host's snuff bottle.

You could also fly to the Gobi for a few days to see the Flaming Cliffs (where Roy Chapman found the dinosaur eggs). Or fly up to Lake Hovsgol to visit reinheer herders.

UB itself is a depressing crowded Soviet-style city, but there are a few monasteries of interest in the outskirts and the people are friendly. Khan Brau (the local beer) is among the worst I've ever had, alas. The State Department Store is a good place to do souvenir shopping as you can buy traditional clothing there.