Sorry to be the bearer of bad news, but my french wife was refused tourist visa at russian embassy in ulaan baator in may 2018. We spoke with another group of french and dutch who were also rejected in march or april. Only canadians, czechs, and the other exempt countries can on the list below can get the tourist visa abroad.
Australia, Argentina, Belize, Bolivia, Bosnia and Herzegovina, Botswana, Brazil, Brunei, Burundi, Gabon, Guyana, Gambia, Guatemala, Guinea-Bissau, Honduras, Georgia, Denmark, Egypt, Zimbabwe, Israel, Indonesia, Iceland, Italy, Yemen, Cape Verde, Cambodia, Canada, Qatar, Kenya, Colombia, Congo (Brazzaville), Congo (Kinshasa), Costa Rica, Côte d'Ivoire, Cuba, Kuwait, Laos, Lebanon, Mauritius, Mauritania, Madagascar, Malawi, Malaysia, Maldives, Malta, Morocco, Mexico, Mozambique, Myanmar, Namibia, Nicaragua, New Zealand, United Arab Emirates, Panama, Republic of Korea, Romania, Salvador, Sao Tome and Principe, Swaziland, Seychelles, Senegal, Serbia, Singapore, Sudan, Suriname, USA, Sierra Leone, Taiwan, Thailand, Tanzania, Trinidad and Tobago, Tunisia, Turkmenistan, Uganda, Uruguay, the Philippines, the Czech Republic, Chile, Ecuador, South Africa, Jamaica.https://mongolia.mid.ru/en_GB/consular
You are probably either going to have to fly back home to get tourist visa, or somehow work with 10-day transit visas - which you can get everywhere. Not great for cycling i know