There are few areas at the Pit that you may be able to catch some shade in the am, and later in the pm, but not many options. The brunt of the Pit faces south, with a few routes that face west. Maybe Swiss tower's right side in the late afternoon. And there are a few in the Bedrock area that will have shade. At Paradise Forks you will be able to find shade, one arm runs pretty much north south, which will allow you to switch sides from morning to afternoon. Davidson wall faces west, and Pillow and Obscurity face east. Good luck
This is a great time of year to climb around Flagstaff. The monsoon season has not started and there's little humidity. PF will be very nice. Davidson Wall in the am and Prow/Pillow wall in the afternoon. The Pit is excellent if you go after 2pm when it's shaded. After 2pm you have great options at the Pit contrary to what the previous poster stated.
We climbed at the Pit on Sunday, and were in the shade all day. Started out in the Oven, then over to Lost in the Woods. By the time we came back, most of Swiss Tower and Mall Wall were fully shaded. I think the right side of the Nose and over by the Abyss would get you some morning shade options as well.
We were pleasantly surprised at the lack of crowds -- hardly anyone out there at all.
Another option for the Forks is the Jungle Wall -- not nearly as many routes as other sections, but north facing.