I read about a sacred mountain in Japan that was not Fuji. You know - the kind with monasteries all around and nice hiking in the hills. So now I wonder if this is a common concept in Japan? And what are the best options to experience this out of Tokyo or Osaka for a day trip or one night stay? I also wonder if the religions are mixed (I guess Shinto and Buddhism) or if there is a difference in religion when it comes to sacred mountains.
Shinto is at heart an animist religion that imbues many natural features (rivers, trees, mountains) with spirits, and Japanese buddhism has been influenced heavily by this, so yes, "sacred mountains" are pretty common in Japan. For Buddhism, sacredness is usually centered around places where famous Buddhist teachers lived, taught or are buried. The two religions are very much mixed, and it is common to have small Shinto shrines within a larger Buddhist temple, and vice versa.
One famous sacred mountain (group of, actually) is Mount Kōya, reachable from Osaka by a 1.5 hour (very sceninc) train ride. It's actually a small town next to the burial place of the founder of an important Buddhist sect. The town has a large number of temples, many of which have rooms for guests to stay in (basically like hotels), and I very much recommend staying for at least one night. The area around the town has some nice hiking trails, but don't expect alpine conditions or grand panoramic views (the mountains are not high enough for that). If you want a bit of a challenge, you can walk the old pilgrims' trail, which is over 20km long (but only 600 meters of altitude difference). You could do this to get there (instead of the final leg of the train ride), but plan for 7 hours walking time.
Out of Tokyo, there is Nikkō, not really a sacred mountain as such, but a mountainous area which has a number of important shrines and temples (including the burial place of the founder of the Tokugawa Shogunate), and lies next to a national park with great scenery (including a lake and a famous waterfall) and hiking trails. Scenery-wise, this is probably the better choice.
Be aware that both of these are major tourist destinations - you won't be alone (except on the hiking trails; the Japanese don't seem to be much into hiking).
If you want to be immersed in Shinto religion, I'd recommend visiting Kyoto. There's lot of temples and shrines to see, and lots to do and even more to eat.
Spending just a day there wouldn't do the city justice, stay a night or two in a traditional Japanese Ryokan and enjoy the scenery. Don't worry about speaking Japanese, there are plenty of professional English speaking tour guides that will be willing to show you around.