Tateyama is probably the easiest mountain to reach in the Japan Northern Alps by public transportation thanks to Tateyama-Kurobe Alpine Route which connects the east and west bases of the mountain (in Nagano Prefecture and Toyama Prefecture) by a series of public transportation lines, so that would be my recommendation.
As for a campground, there is one called Raicho-zawa Campground located near one of the highest stops on the Alpine Route on the Toyama Prefecture side, which sounds like it is what you are looking for. Though information appears to be available only in Japanese, I will translate the main points of the above link:
- Located near Murodo (室堂), a bus stop on the Alpine Route that can be accessed by public transportation from either Toyama Station (coming from Kyoto, Osaka) or Shinano-Omachi Station in Nagano (coming from Matsumoto, Tokyo).
- The campground is about a 30 minute walk from where the bus drops you off.
- For one night it costs ¥500 per person, for staying two or more nights (no matter the number) it is ¥1000 per person.
- The pictures of the campground shown on the webpage linked above were taken in August and September, other than the very bottom picture (with lots of snow) which was taken in November. August should be a very pleasant time of year to stay there.
- There is running water and toilets.
- There is an onsen lodge about halfway between the campsite and Murodo Station. You can use the bath there for ¥700 even if you are not staying at the lodge.
- Many mountain hikers and climbers use this campground as a base of operation for exploring the Tateyama area, as well as those venturing further up/down the Northern Japanese Alps.
The Tateyama-Kurobe Alpine Route generally only runs between 9am and 4pm, so make sure the day that you make the journey up to Murodo you allow plenty of time, particularly if coming from Matsumoto Station (the Nagano side, which is closer to Tokyo). I would say leave Matsumoto around 8am.
The Alpine Route itself is pretty interesting, here is some further information about it in English.
Note: I have not been to the campground myself, but have taken the Alpine Route and soaked in the onsen up there, and can say it would be good area this time of year for hiking.