That's a cute patio!
Since you're in California where water is both expensive and scarce, I'd recommend drought tolerant plants. Do you have access to an outdoor spigot? Who would be paying for this water? Is your landlord okay with you planting somethings there? Even something permanent like trees? Also, trees are expensive! Are you okay with spending that much money in landscaping that you can't take with you when you move?
Also, two feet wide isn't very much space for tree roots, especially in drought-stressed LA. And surrounded by all that hardscape, that patio's going to get really hot.
I like Rosie's idea of a climber. Vines can spread quickly so they can be cheap to start, and I'm sure there are drought tolerant native vines. How about a rosemary that has a spreading habit (so not shrubby)? It would drape over the planter and provide a green backdrop. And they're very drought tolerant!
