The song played at Tatiana's Naming Day feast is "On the Hills of Manchuria". It is not a folk song. It was written in 1906 by Ilya Shatrov, bandmaster of 214 Mokshansky infantry regiment, after tragical events of Russian-Japanese war.
Toby Stephens and Ralph Fiennes would both later go on to star in the James Bond films. Toby Stephens would play Gustav Graves in "Die Another Day" and Ralph Fiennes would go on to play Gareth Mallory/M in "Skyfall" and "Spectre".
The trivia item below may give away important plot points.
Much like the character of Lensky, author Alexander Pushkin was killed in a duel after challenging a man for seducing his wife, Natalya. Pushkin's home in St Petersburg can be seen in the background when Onegin sits on his balcony in this film's final scene.