I've been playing this. I've printed out the version on Wikifonia and the piano version of Borodin's quartet. I'm convinced that whoever wrote it down made a common mistake - put the pick-up note as the first note of the song.

Try making that first quarter note a pick-up note. Then you will see the chord changes happening at the the beginning of the measures and the weak notes occurring at the end of the measure, as is usual.