As I look at that picture, a couple of things are evident.
First, there is a tremendous amount of thermal mass to capture heat and hold it. It's not just the sun that is making that place hot, but the bricks and
concrete hold onto that heat and make it hotter and hotter as the day goes on. The air in the "well" is trapped and the temp continues to rise. I would imagine that with that much thermal mass, it would stay warm late into the evening. You can probably go out there at midnight and the walls will still be warm to the touch.
Second, where would cool air even come from? Any air moving into that space would likely have to pass over the roof-tops of the buildings that surround it, and then drop down into the "well" that is created by the walls of the buildings. So the air will already be pre-heated even before it drops down into the well. Ever walk on a flat roof? Ever walk on a flat roof that has some form of black-tar roofing? It'll be 120 or more up on a roof like that. That is where your "cool" air is coming from.
Third, there doesn't appear to be any sort of cross-ventilation. That space creates a pocket of air that isn't being exchanged. Think of a similarly sized field without walls around it. How long would it take for the air on that field to be exchanged? Maybe 5 seconds? Just a little breeze and all the air quickly exchanges. But in the space you've shown us, the air sits stagnant and isn't exchanged.
In classic "the problem is the solution"
permaculture, I'd pick plants that thrive in such intolerable conditions. Tomatoes and peppers come to mind. But I'd look for a way to increase the humidity of the space. Perhaps you could lay a long stripe of plastic down in a shallow
swale that runs the length of the courtyard and fill it with water. That way you'd hopefully increase the moisture/humidity in the air. Tomatoes like hot muggy air. But I'd stay away from plants that thrive in cool breezy locations: lettuce, brassicas, etc. The west and south walls will get a bit of shadow during the hottest part of the day. You might be able to plant some stuff in those locations where they will not get baked by the hot sun. I'd plant heat loving nut trees like almonds or pecans. In 10 years, that will help bring down the ambient temperature by 20 degrees or more.
If you planted about 200 tomato plants and trained them to grow up a
trellis, maybe that would help absorb some of that sun and keep the ground from getting too hot, thus keeping the overall ambient temperature of the space a few degrees cooler. Or if you installed misters (like they use on patios in Palm Springs or in Arizona), that might keep things a bit cooler. But as it stands right now, a crop of carrots, potatoes and beets are going to get scorched in that courtyard.