You can stay at hotel and guesthouses, which are available in the capital city and experience sleeping in gets (traditional nomadic dwelling with facilities and restaurants and any other equipment) at tourist camps in countryside. Rooms of deluxe, semi-deluxe, first class and tourist categories are available at 3 or 4 star hotels in Ulaanbaatar. In the city it can be also offered a choice of staying at budget hotels and inexpensive guesthouses. They have a kitchen and serve breakfast, internet access, laundry. They are cheap and comfortable enough for those who are interested in authentic (or rough) travelling.
You should note that not all hotels can meet your good expectations. For instance, once or twice every summer hot water supply may cease for annual repair works made at the power centrals.