Eastern Mongolia is a region in Mongolia.
There are now paved roads all the way to Baruun Urt and Choibalson. There are still plenty of options for buses to the provincial capitals and daily flights to Choibalsan from UB and Hailar, Inner Mongolia. Airports in Ondorkhaan and Baruun-Urt no longer have scheduled flights to UB. There is a rail line from Choibalsan to Russia, but it has no passenger service.
First you need to find a reliable driver and negotiate a price. Getting to a province in Mongolia from the capital is much easier than going between provinces. The best strategy is starting as far from UB as you wish to go, say Choibalsan, and working your way back to Ulaanbaatar. There are shared jeeps between provincial capitals and from the cities to the countryside.
The main attraction of Sukhbaatar Aimag is the steppe, which is flatter and truer steppe in this part of the country. Also be sure to look out for white tailed gazelle on your journey.
Drive around, go camping, sleep in a ger.
Go to Dariganga Soum and visit Altan Owoo (a sacred hill). Women can only go the foot of the hill. Please be respectful.
Related Wikipedia article: Dornod Province
Related Wikipedia article: Sükhbaatar Province
Related Wikipedia article: Khentii Province