You can get to Mongolia by a plane or on train 

Air Gateway to Mongolia

-   Flights from Moscow to Ulaanbaatar by Aeroflot /Russian Airlines/ and MIAT /Mongolian Airlines/
-    Flight from Irkutsk to Ulaanbaatar by Aero Mongolia /Mongolian Airlines/
-    Flight from Berlin to Ulaanbaatar by MIAT
-    Flight from Istanbul /Turkey/ to Ulaanbaatar by Turkish Airlines
-    Flights from Beijing to Ulaanbaatar by MIAT and Air China /China Airlines/
-   Flights from Seoul to Ulaanbaatar by MIAT and Korean Air /Korea Airlines/ Asiana
-    Flights from Tokyo and Osaka to Ulaanbaatar by MIAT
-    Flights from Hong Kong /China/ to Ulaanbaatar by MIAT 
-    Flight from Nur-Sultan /Kazakhstan/ to Ulaanbaatar by SCAT 


Trains to Mongolia /Trans-Siberian train/  

Train from Moscow to Ulaanbaatar takes 5 days to get to Ulaanbaatar, Capital city of Mongolia. 
Train from Beijing to Ulaanbaatar takes 28 hours to get to Ulaanbaatar, capital city of Mongolia.