As others have pointed out, the song was originally by singer "Fats" Domino, and so '"Fats" Putin' is a reference to this.
Which, to answer the question, makes "Fats" a nickname (or as a part of speech, a proper noun).
And because I've just realised I missed the second half of your question...
To sing "over the top of" something usually means a karaoke-style situation: there's a performance (more often recorded, but in this video live), and you're singing it too... but your voice is being heard "over the top of" it, because you've got a microphone or just a loud voice.
Now, normally you wouldn't say that (for instance) the lead singer of a band is singing "over the top of" the backup singers and the musicians; they're part of the performance. But in this video, it's an appropriate description, because Putin isn't really part of the band. He's there because it's a charity event where famous people do unusual things like this. So it's fair to say the band is performing so that Putin has something to sing "over the top of".