Fight Russian Corruption Act This bill amends the State Department Basic Authorities Act of 1956 to require the Department of State to establish the Office of Anti-Corruption relating to Illicit Russian Financial Activities in Europe. Such office shall: (1) analyze financial networks of the Russian Federation operating in European countries that relate to real estate, energy, media, infrastructure, and other sectors; and (2) train U.S. liaison officers to serve in key U.S. diplomatic and consular posts in such countries to cooperate with foreign partners in uncovering and prosecuting illicit Russian financial activity. The bill authorizes the State Department to provide assistance to European countries to combat corruption, including to support: (1) activities of such office; (2) foreign programs focused on investigative journalism and independence of the media environment to expose Russian corruption; and (3) activities in Europe related to anti-corruption, anti-propoganda, and anti-Russian malign influence. The State Department shall seek to work with the North Atlantic Treaty Organization (NATO) to: (1) elevate anti-corruption as an element of NATO's Readiness Action Plan, (2) task the NATO Assistant Secretary General for Intelligence and Warning with monitoring Russian influence in NATO member states, and (3) prioritize the combating of Russian influence under the NATO-European Union framework. The Office of the Director of National Intelligence shall submit a National Intelligence Estimate on: (1) purchases made in the last five years by individuals and entities of key sectors in European countries, particularly purchases that provide monopolistic control of a sector; and (2) the individuals and entities making such purchases, including any links to the Russian Federation.