Russia and the United States are both parties to the Hague Service Convention, but the Convention doesn't work between them in practice. In July 2003 Russia suspended judicial cooperation with the United States in civil and commercial matters, and according to the U.S. Department of State, Hague requests sent to Russia's Central Authority and Letters Rogatory sent through diplomatic channels are returned unexecuted.
That leaves U.S. litigants with no reliable official channel. Russia has also objected to the alternative methods in Article 10. U.S. courts have split on what follows: some have authorized alternative service, such as email, under FRCP Rule 4(f)(3), while others have held that the Convention still rules those methods out. Any plan to serve a defendant in Russia should be built with counsel around your particular court's decisions, and with U.S. sanctions in mind.