Maybe add some silent space with an audio editor before the music starts, negative offset is probably a bit too soon. but if you aren't going to map that beginning at all, try 5,320, or alternatively 1,320 to at least get three of those first 4 measures. Also use 1/3 snap, the song is swing, meaning instead of even eighths (1/2) the song is broke up into triplets (1/3), usually with emphasis on the last triplet of the beat (second purple tick).