Julia,
My limited experience is that the selection of boots and shoes is pretty decent in Ulaanbaatar, though if you have larger feet the selection is not so good.
It probably wouldn't make a big difference either way, but given you plan to arrive in January you might be better off buying a pair you like in Europe and bringing them to Mongolia. You will probably be busy enough with other things when you arrive, and it's no fun walking around shopping in light footwear in January weather. You also might not have the full variety of brand names that you're used to in Europe, if you're selective about such things.