Yeah, 2nd week of June should be at least a week before rainy season ("tsuyu") starts. Still, some days going to be rainy for sure.
There's quite a lot of places to go to around Kyoto/Kobe/Osaka, sport, trad, and bouldering, but they might be quite hot: expect temperatures around 30 degrees Celsius. Horai is really good for sport and hard bouldering (a few problems up to 7a, but most stuff is harder than that). Ogawayama is not a bad idea at all. The campsite is at 1600 m above sea so it should be a lot colder, and I'm sure its bloody cold during nights. I takes something like 5+ hours to get there from Kyoto: just get up on the chuo express-way and drive all the way to Sutama-intersection (須玉IC), then the fun starts....
So there is a English web-page on ogawayama.com I have written some stuff on ukclimbing:
http://www.ukclimbing.com/articles/page.php?id=264 I found a problem with the map I drew. Where it says "Bypass" you have to look out for a sign pointing toward 廻り目平 ( <--- sorry if you have an old OS: then you only see: ????) Its a very sharp fork and can easily be missed.
I really, really liked the boulders in around Mizugaki: put up some info here:
http://thousand-cranes.blogspot.com/2007/04/mizugaki.html It takes something like 25 min to drive from the campsite in Ogawayama to Mizugaki forest. If the weather is still crap tomorrow I'm going to add info how to get to Mizugaki from Ogawayama. A map of sorts can be found in Mr. Kitayama's topo.
If you get fed up with trying to find the routes you want to climb in the confusing jumble of rocks and spires on the forrested hills of Ogawayama, you can always go to the small trad-crag Yukawa. Half an hour by car from the campsite.
If you don't want to drive (coward...) you can rent a room at the Ryokan at the campsite. A friend of mine did this last year, and they came and picked her up at the trainstation. I stayed at the ryokan October 2005, and they had excellent food (breakfast and dinner) in a very traditional Japanese style. It might be a good idea to book soon, I think its quite popular in summer.