While the western side of the Castle Mountains is mostly granite, the eastern side tends to display limestone.
Much of the limestone (and granite) in the Castles is choss, but in Green Canyon, there is limestone that is both climbable and exceptionally beautiful.
For better or worse, the information on the Mountain Project page is sparse, and I actually wasted a little over an hour locating the routes I wanted to climb. Alas, this page will be little or no better, though I hope my commentary may help the few who read this and enture out.
It seems that by far the best source for information is The Big Empty by Ron Brunckhorst.
For the driving approach, I am going to copy from the page linked above and give due credit to the author, Adam Pankratz, because his directions are excellent and it would be a waste of my time to rephrase them:
"East of White Sulphur Springs on US-12 find Pasture Gulch Road between mile marker 68 and 69. Head south along a Forrest Service road through private property until you reach a Y intersection, head left to the east on Pasture Gulch. Shortly there after you will reach a second Y. Head left again switch backing to the north then south. You will come to a USFS access gate that is locked from Sept 1 to Dec 1 during MT's long hunting seasons. From here drive or walk (3) miles south to a wide right turn in the road. You will see a two-track trail heading up hill to the SE through a clear cut. Park here and begin the (1) mile walk to the crag. As you crest the hill you will see the limestone below through the trees to the NE. Stay on the 2-track as it continues down hill to the SE then turns NE. It will follow a small stream for a short ways. The 2-track will take you directly to the rock."
My picture below should help if there is any confusion about where to park.
Because online information is so lacking, I will list the crags I know of and describe their general locations.
Welcome Rock-- This is the first bolted crag you reach and is on your left, and a picture on this page shows it. The MP page documents a 5.4 sport route called Welcome to Green Canyon on it, but I did not see the bolts when I passed by.
Ponderosa Wall-- This is just past Welcome Rock and also on the left, with the wall facing the trail. Although the MP page mentions nine sport routes from 5.7 to 5.12a, only two route pages exist right now. One is for The OK Corral (5.7), and I have a picture of the route here. The other is for The Big Empty (5.8+). Note that the route pictured on MP (also pictured here) does not match the desription in some important respects, namely being that the description mentions a large crack to the left throught the lower section and there is no such crack on the route pictured. Furthermore, if the pictured route is the correct one, it is the hardest 5.8 I have ever done by far. I was lead-soloing, which always makes me more timid and makes things seem a little harder, but I found some lower sections pretty blank and more like 5.11 or a hard 5.10, and I aided my way through some moves to avoid falling. I think the second route pictured below may be The Big Empty instead.
Update: I have learned that the slabby route in the third picture is a 5.11 called Come Full Circle and that the Big Empty is the next route to its right.
A climbers trail leads up along the base.
Chertstone Tower-- This one is the next bolted crag after Ponderosa Wall. From one of the MP pages, I inferred that both crags are part of one formation called Chertstone Dome. MP includes one route, Pecan Clusters (5.8-), but mentions that it has four routes, including Green Canyon's hardest, The Honeymooners (5.12b).
The Corral-- All I know about this wall is that the MP page for Chertstone Tower mentions that CT is between Ponderosa Wall and the Corral.
None other than respecting posted private property.
Spring through fall.