By car
The road is paved all the way, fine for cars. Lots of UB residents come for the weekend, so you can probably find a ride back to UB pretty easily. Offer to help pay for petrol.
By bus
There is a bus from Ulaanbaatar to the park.
It leaves from a bus stop on the south side of Peace Avenue, opposite of the big 'Narantuul Hotel', 150 meters East of the junction with Ondor Gegeen Zanabazaryn Gudamj, which is the road that runs south from Gandan Khiid. The bus waits at the east end of the bus stop. The bus stand has a large "e-ticket.mn" sign on it GPS N47.91545 E106.89782.
Buses leave at 16:00pm, and take 2 1/2 hours to get to Terelj village.
Coming back, they leave Terelj village at the uncivilised hours of 08:00am and 19:00pm. The bus costs 2300 MNT one way as of July 2009.