iagospear-
My wife already owned an apartment in UB so we've never had to search for rentals, and I'm not really that familiar with the rental market. One thing I can say is that you probably want to visit any apartment you are interested in IN PERSON and do not commit to any deal before you arrive and get to look at the place. There are big differences between various apartment buildings, and even if the interior pictures look nice, the exterior and the rest of the building can be a lot different. My apartment is like this... inside the apartment is pretty decent, but the hallways and building exterior are in serious need of repair and remodeling. Location is a big factor too... ideally you want to be close to your workplace and close to grocery stores and shopping areas.
You CAN drive if you really want to, but be warned that the traffic can be extremely bad and driving here can be challenging. And if you're actually living here for an extended period and going to be driving you will need to get a Mongolian Drivers License, which also can be a bureaucratic challenge and take a long time to get (took me about 3 weeks to get mine). Long story short, unless your job requires you to drive, you are probably better off finding an apartment close to work and walking or taking taxis (or the bus) where you need to go. We do have a car ourselves, but we still walk more often than we drive.
Here's a couple websites about renting in UB or showing apartments for rent (I don't use the sites or businesses myself so I don't know how good they are, but it might help you a bit and give you an idea of the prices):
http://degruben.com/how-to/913/how-to-r … aanbaatar/http://www.mondinion.com/Apartments_for … /Mongolia/
Oh and one more thing, apparently around here some places say they are a 2 bedroom apartment, but what they really mean is that the living room is counted as a bedroom, so there is only one "separate" bedroom.
Hope that helps.