Probably not what you are after, but there is something called:
"Kije! A musical fairy tale" out there. I fear it is too recent to be connected to your school tune.

What do you recall about the tune of your song?

The most famous name associated with Lieutenant Kije is that of Soviet/Russian composer Sergei Prokofiev, whose film score for a Kije film was adapted into a suite for symphony orchestra. Prokofiev's Kije Suite is much recorded, a staple of the classical symphonic repertoire. No songs or lyrics, though.