Wake up early and take advantage of the only cool part of the day (before 9am–it will not cool off in the evening). Do this when you’re up in the high country too. Park your car in the shade, even if you have to walk a bit farther. Take water with you everywhere and drink it. I’ll add another vote for the Musical Instrument Museum. Sports? The Diamondbacks may be playing at home this weekend and Chase field is air conditioned.
There’s lots to do up in Flagstaff, especially if you enjoy hiking, but still wake up early and enjoy the cooler mornings. You can always nap in the afternoons. Lowell Observatory is nice if the weather is clear. There’s often afternoon and evening thunderstorms in the summer.