What kinds of plants are you thinking of planting. Herbs?
Pots dry out very quickly here. Herbs like shade and would do good in shaded areas. Just remember with pots and this heat even shaded you might need to water every day.
I would go with raised beds a least a foot deep. Then if you want you maybe could throw in a small bush squash or some radishes, lettuce, korabie, strawberries, garlic and onions or even some flowers. Question- are you really sure about the amount of sun, that sound just lacking for this area.

Shade tolerant: Alpine strawberries (year round producer, perennial)
Arugula
Basil
Celery
Kale
Lettuce
Peas
Potatoes
Spinach

We get so much reflected light and it is intense around here that 2 hours of light is likely equivalent to 3-4 hours almost anywhere else.

That said, June is the worst month to plant much anyway.
http://www.phoenixpermaculture.org/forum/topics/the-100s-are-here

Keep watered and mulch.

Here is a link that might be useful: shade tolerant

This post was edited by Fascist_Nation on Mon, Jun 2, 14 at 21:27

thank you for the follow up advice! We just moved in but it's a smaller patio that is enclosed with a wall and just outside the wall is a huge tree so unfortunately I haven't seen it get more than 2 hours of partial shade.

I'd love to grow any kinds of edibles. Probably an herb or two and some vegetables would be great! I can wait to plant and research further, I just didn't want to build the raised bed if nothing would grow there.

Would I put the mulch surrounding the wood of the raised bed? or on top of the soil? Also which type of soil would you recommend? I have an Arizona planting calendar I can follow for the recommendations of plants y'all have provided.

Again, thank you for responding! There is so much information out there.