The road from Ulaanbaatar to Terelj village, which goes through part of the national park, is in pretty good nick. A bus (T2500, 2½ hours) departs daily at 4pm, from a stop opposite the Narantuul Hotel. It then stops at every bus stop heading east along Peace Ave. Get there with 15 minutes to spare as they sometimes depart early. Going the other way, it leaves Terelj at 8am Monday to Friday and 11am on weekends; catch it near Hotel Terelj. You may have to pay an extra T2500 for a bike or heavy luggage.
Another option is to take one of the frequent buses between Ulaanbaatar and Nalaikh (T900, every 10 minutes), and then hook up with one of the shared minibuses that shuttle between Nalaikh and Terelj village (T1500, roughly every hour). A taxi to Terelj from Nalaikh costs about T25,000.
A taxi from Ulaanbaatar (T70,000 one way) is easy to organise; jeeps aren’t necessary because the road is paved all the way. The driver may understandably want more than the one-way fare because their taxi may be empty for part of the return journey. You can also arrange with your taxi to pick you up later.
When there’s enough demand, shared taxis to Terelj sometimes leave from Naran Tuul Market jeep station in Ulaanbaatar. This is more likely on summer Sundays when locals make a day trip to the area.