I would recommend two places the sky lounge restaurant for it's view on central Ulaanbaatar, this will give you a good idea of how the city is built and what to expect from it. That restaurant is far from being cheap, but the food is just delicious - so if you are on a budget I would not go there. You'll need to be dressed a bit to be allowed to walk into the restaurant, but as said, well worth the view and well worth the food. If eating isn't your thing then just enjoy their lounge bar.
If you are more on a budget I would recommend going to seoul street and try either the American restaurant or bd mongolian barbeque - the later being a wok like restaurant with tasty food and if you're lucky a nice show from the cook while he prepares your food.