The example on the video is very well done and has probably been rehearsed.
You don't have the luxury of rehearsals so you can't expect such a polished performance.
Some songs have traditions: for example often verse is rubato, chorus is in tempo. If you're not sure you can ask the singer beforehand.
I find that doing rubato on a song that you're sight reading and never heard before is the hardest.
When I have to do that, then I read the words that the singer is singing, and I just follow along and fill the holes with little arpeggios. It's better to be a little behind the singing that ahead. Also, if a phrase starts before beat 1 of a bar, then you wait for the singer to start and play the chord when he reaches beat 1. But, if a phrase starts after beat 1 then it's on you to play a chord (usually the singer will be waiting to hear that chord).
Always be looking at the singer (and at the sheet with your other pair of eyes
) because often you can pick up cues as to what's coming. Be flexible! Sorry that I can't think of anything more useful...
Another problem that you might run into is that they might want to sing in keys other than the sheet music that you have. This can be pretty hard to pull off on a song you don't know...