Mongolia isn't flush with options for escape. If you want to fly, you'll be flying from Ulan Bator (ULN). Your main choices are to Beijing on Air China or MIAT, or to Seoul on Korean. These are not cheap flights and especially not in the summer high season. I would expect just this segment alone to be at least US$250 one-way. First half of July is worst with people coming/going to Nadaam Festival. There are a couple of "back door" routes into China(Aeromongolia ULN to Hohhot and Eznis ULN to Hailar) but these aren't daily and aren't going to save you money.

From Beijing you can fly nonstop to Vietnam on VN which is a pretty decent flight. From Seoul on KE (maybe OZ as well). It's almost certainly going to be cheaper from Beijing. I've done this segment PEK-HAN one-way before for about $250 but that was a couple of years ago and prices are surely up. (purchased from a travel agent in Beijing not online).

To save some money, you can also consider international train (overnight, sleeper) from Ulan Bator to Beijing, then fly from there. Another method is to fly from Beijing to Nanning then train or bus to Hanoi from there--the two cities are very close.

If you are in Mongolia now, you should check with a local travel agent there, also with MIAT directly, at least to get as far as Beijing. These sorts of countries often using the old-fashioned TA method is better than online.