First of all, please check that do you need visa when you enter to Mongolia. 
List Of Countries That Require No Visa, List Of Countries Without Visa Fee

Practically a 30-days tourist visa can be obtained from any Mongolian Embassy abroad within business 5 days for approximately US$ 40.00. Additional fees are charged for urgent visa.
Addresses Of Mongolian Embassies, Consulates And Missions Abroad

Alternatively, those who travel from countries with no Mongolian mission present can get visa either from the Mongolian Embassy in Beijing or Moscow,
How To Get Mongolian Visa In Beijing?

or at the arrival points: “Chinggis Khaan” International Airport in Ulaanbaatar, “Zamyn Uud” border point or “Sukhbaatar” border point.
How to Get Mongolian visa on arrival points? 

Visitors staying longer than 30 days are required to apply for registration with the Office of Immigration, Naturalization and Foreign Citizens. Visitors planning to stay for more than 180 days should obtain long-term residence permission. Please beware that that procrastination with visa extension may result in several visits to the Ministry of External Relations and up to $ 5.00 fine for each day after visa has expired. Visitors are also advised to obtain proper transit visa from China or Russia. There were cases when foreigners who failed to secure two-time transit visa were denied entrance to these countries on their way back home from Mongolia.

NOTE: USA citizens can travel now to Mongolia without visa, as a respective agreement was signed in 2001.