You must have a complete knowledge of your arranger and know where every style resides and what it sounds like. Of course this is not totally possible but you can get close.
I make a point to try to do every request unless it is not suitable for my abilities or taste. If I can't do it and want to do it I will promise to learn it for the next time they come.
Sometimes I will play an Mp3 of it when I take a break, if the song is appropriate for the venue.
One suggestion-never apologize if you try it and don't think it is perfect. Normally, they will hear the tag or chorus, identify with it and be happy. They aren't paying THAT much attention.
I do have generic Songbook entries for most style setups. They include slow 6/8, slow 4\4, 2/4 country, shuffle, Latin, swing, big band, rock & roll, etc.