I've converted many an apartment patio to unconventional purposes!! If you are reasonably handy...
I'm assuming it's a standard size apt. patio, and you are willing to devote most of it to the hens?
It looks like you have already mounted some type of wood frame against the stucco wall you can use as a base. If not, I'd use some anchor bolts, and put up a couple of pieces of 2x4 against the stucco to make a framing. Put your little coop inside the frame. Cover the roof and one end (the end opposite from where your apartment door is), with hardware cloth. Those apartment living cats are stone-cold killers!! Build a simple little 4' frame door in the end closest to your door. If you are really building impaired, you might be able to make do with some odds and ends to put together a door - like a fireplace screen, or some large venting. 4' is plenty of room for you to get in to be with the hens, although taller is always better, it looks like you need to stay below a certain level.
Get a couple of bags of gravel for the flooring and plan on scooping. It will be much easier to deal with than shavings or straw in an apartment, and stays tidier when you (or the hens) are spilling their water!