Go to Terelj nature reserve or to Mandzhir monastery. For only a day stay go to Sainhand war memorial, have a nice look to UB and on the way back visit the Bogd Gegen Museum. This museum is the residency of the state chiefs and the former leaders of Mongolia.
, 1 Week,
Visit Lake Hovsgol, it is 24 hours (direct way) or 2 days by public transport.
Make the 10 day trip with a group to Gobi desert, it is really worth seeing it.
The round trip: Lake Hovsgol, Bus trip to Uliastai, Hovd, stay a week in Bayan Olgii - mountain climbing, visit the Tsengel Sum..... back to Hovd, Altai Hot, Bayanhongor, UB (rather 6 than 4 weeks) needed
longer go to the Heartford institute near London, make this 500 Euro english teacher exam. This entitles you to teach in UB at the Heartford institute or anywhere else in the country. Also they have settlements all over the world, so a work permit is not a problem. Actually work permits are only agreed to specialists - language teachers included. In UB they salary is 500 US$ a month, this is enough to survive and rent a room for a longer time or to pay the tuition fee at UB university for mongolian courses at university level.
US citizens can become peace corps members and stay one or 2 years in Mongolia. Or go as a volunteer and help local families for various periods.