We take this part of the route 6 almost everyday. I know it so well I can drive there eyes closed. But what a surprise, when I tried doing this recently the eyes open, I spotted a totally new sign saying "Nuchishinujigama cave"!
So, it looks like the cave is on private grounds. The GZ coords lead you to the cave 'office' (opposite corner from the cache), where you can get a brochure and pay the entrance fee. It's 100 yen/person for walking down the cave and seeing the entrance hall, and 1000 yen for going inside with a torch. Looks like crookery, but the guy built stairs and ladders at some places, and he's kind of maintaining the whole place which also shelters some sort of tomb or sanctuary. So far we haven't explored the cave, just seen the entrance hall, and it's kinda promising. I haven't put the cache closer to the cave because I was not quite ready to ask permission for this in Japanese.
The cache is accessible all the time, but the cave is probably not (well, it is, but the 'office' would be closed).
FTF gets +1 luck and +1 moral for the rest of the week.