There is a reliable mobile phone shop on the second floor of the State Department Store, it is operated by Mobicom which is the largest operator in the country. They have a good level of coverage and are relatively inexpensive. It is recommended that you go with a Mongolian friend who can help you fill out forms and make the right choices. Prices very much depend on the packages you take but are no more expensive then elsewhere in Asia.
 
It's advised if you're in Mongolia for a long period of time (two months or more) to buy a Mobicom, or Skytel Sim-Card. They come at an affordable price, and are useful for contacting your friends, and in an emergency. Just don't forget to top it up! 
 

 
Phone calls and  text messages are almost negligible in price, and a ten dollar top up should last you a month or so.
 
There is a large Mobicom centre, about a five minute walk behind the State Department store. It's ideal for buying new phones, chargers, batteries, and all other types of paraphernalia, with extremely reasonable prices.
 
Most shops as well as the post office, the tourist information office and the state department store will be happy to sell phone cards to call home to use on the payphone.