For all practical purposes, the Mongolian tögrög is a closed currency, meaning you can't buy it outside of Mongolia. I'm not quite certain why this is so, but probably because it seems to be illegal to take more tögrög out of the country than you bring in, so there's no easy way for an overseas bank to gain a supply of tögrög to sell to you. (But don't worry about getting into trouble taking a few left-over notes out of the country.)
ATMs are plentiful in central Ulaanbaatar. They're often inside banks, rather than on outside walls, and are typically labelled 'ATM' or 'банкомат'. I had no difficulty using a British bank card there. If your first stop in Mongolia is Ulaanbaatar, you won't have any difficulty obtaining currency. If you arrive by train, then it's normal for a man to come down the train selling tögrög in exchange for various currencies. I brought some US dollars with me for that purpose, as that tends to be the most easily traded currency.