How to best learn music fast

Some weeks ago, in an attack of hubris, I said yes to joining a big band as their new guitarist. It has been a while since I had to read music. Honestly, I have, since a short theatre gig in 1986, not had to have paper in front of me much while playing. I know all the 1845 chords that need to be handled in a gig with the big band, but there are passages, for instance when Cherokee is played at 300 km/h, that I have to learn by heart. My preferred method of rehearsing this is to copy sound files (they gave me the sheets and mp3 of all songs) into Ableton Live, and loop especially difficult passages in order to play them over and over again until I can play along. The sequence is: first reading the passage, singing and playing it slowly, then turning to Live to play along. It works in a way, but I feel there might be a better way.