That's the Tucson Mountain Park range, nice little place.
There is the Marksman indoor ranges, one on 29th and the other by I-10 on Prince/Flowing Wells. The last time I looked, there is nothing by Catalina Foothills, and I grew up there.
You will have to hike out a ways. I don't know if Desert Trails is still open waaaay down on Kolb Road, that was an interesting place.
If you want to go way out there for a neat membership range, Tucson Rifle Club is out by Three Points, has many different ranges including a 1000 yard range.
Up here by CG we have a GREAT free outdoor range, but a long ways to hike unless you want to stop by for barbeque.
Edit to add, I forgot the Pima County range down by the fairgrounds south of town, but I never went there, and know next to nothing about it.