I am wondering about the availability of public transport to Mutiyanyu.

Anu, you can take no. 916 bus (express) from Dongzhimen and get off at Huairou North Street. Then walk to the other side of the road to take bus H23, H24 or 936 (Huairou to Dongtai) and get off at Mutianyu Roundabout. There you can take the shuttle bus to the ticket office. If you don't want to take the shuttle bus, you need to walk around 3 kilometers.