On a Russian site, there was a seemingly snubbing remark about how they were singing to "Kalinka(-Malinka)", the title of which is often confused with "Korobushka". The meter of the chorus given above doesn't fit "Korobushka", while it fits Kalinka's chorus if you force it a little. Of course, they may have been referring to another song in the movie. Other sites mention that much of the scoring borrowed from Ukrainian music, so it may be a Ukrainian song. Sadly, the soundtrack (reissued on CD) doesn't appear to include any songs from the film, nor do film databases mention them.

I'll scan my Russian copy of Gogol to see if there are any lyrics in the book that they might have used.

The translation reminds me of a well-known Russian song that is similar, but not close enough that I'd hazard it's the same one. It began in the 18th c. as a students' drinking song (as given below), although it has since morphed into military and "people's" versions.


Наша жизнь коротка

1. Наша жизнь коротка, все уносит с собой,
    Наша юность, друзья, пронесётся стрелой.

Припев:
Проведёмте-ж, друзья, эту ночь веселей,
Пусть студентов семья соберётся тесней.

2. Не любить – погубить значит жизнь молоду.
    В жизни pай – выбирай каждый деву младу.

3. От бессонных ночей голова не болит,
    А болит у того, кто без просыпа спит.

The verses are sung slowly, with a sort of mournful feel; the chorus is livelier, and in some versions expanded thus:

Припев:
Проведёмте-ж, друзья, эту ночь веселей,
И пусть наша семья соберётся тесней.
Налей, налей бокалы полней,
Налей, налей бокалы полней,
И пусть наша семья соберётся тесней.
И пусть наша семья соберётся тесней.

Here's a hasty stab at a translation:

1. Our life is short, everything is transitory [carries itself off]
Our youth, friends, passes by like an arrow.

Chorus:
Let's spend this night merrily, friends,
Let the family of students gather tightly.
[Pour, pour, fill the glasses,
And let our family gather tightly.]

2. Not to love means to destroy one's young life.
In life is paradise; select [take out?] every young lass.

3. Sleepless nights don't cause heads to ache,
Those whose ache are those who sleep all hours.