There is a bus going from UB to Terelj.
If I remember correctly, it runs once every 2 hours? (at least several times a day). We got our info in one of the bars/cafes that is run by an expat. (I don’t remember the name , it is a cafe that kinda stands out. lots of wood, it somewhat resembles a mountian hut. They organise tracks and stuff.) the place is called Chez Bernard Café
The bus leaves somewhere from the main street, and drops you off at the village at the south of Terelj Nation Park.
Credit:stackoverflow.com‘