As of 2013, Turkish Airlines started flying to Ulan Bator directly from Istanbul, or on some days via Bishkek. Depending on the exact days of travel, that might be the cheapest connection from Europe.
Wikipedia is a great source for these type of questions, I always use it as a primary source to find which airlines fly to a specific destination. This enables me to narrow the search for an interesting fare.
To answer this question I have entered “Wikipedia Ulaanbaatar Airport” in Google and I was directed to this Wikipedia entry:
According my search, the Aeroflot has the cheapest price (near $900) for this trip, with 1 stop in Sheremetyevo (SVO):
And I didn’t find any flights in mid August for Air China.
Credit:stackoverflow.com‘