Mongolian Cyrillic uses the same alphabet as Russian, but with two extra characters, Өө and Үү. Unfortunately these are not included as options in any of the Cyrillic keyboard layouts included with OS X, so you need to use the Character Palette or install a custom layout. You can download a MongolianCYR and MongolianQWERTY layouts that do have them here.
Lucida Grande is the only font that comes with OS X that has the two extra characters. If you have Office2004, then the Arial, Monaco, Times, and Times New Roman that come with it should also work. Others you can download are Everson Mono, Charis SIL, Doulos SIL, Code2000, and Gandhari Unicode. Some MS Office Chinese fonts have them, but these are double-width.
Some users have reported that the keyboard works in every app except MS Word. If that is a problem for you, try typing some Mongolian in another app and then copy/pasting into Word. This may force it to recognize the keyboard.