See which operator a number is on?

Is it possible to find out which operator a Japanese mobile number belongs to? Are the 080/090 numbers assigned in blocks to the operators? How does MNP affect this?
I noticed that http://www.jpnumber.com/mobile/ lists Docomo or Softbank Mobile for certain numbers. How reliable is that information? Why is it available for some numbers only?
Is there some form of publicly available access to the MNP database (which should provide the ultimate answer to my question).

Is it possible to find out which operator a Japanese mobile number belongs to?

Not since MNP was introduced.

Are the 080/090 numbers assigned in blocks to the operators? How does MNP affect this?

090 numbers are mostly assigned to Docomo and Softbank while 080 numbers were assigned to au. But after MNP was introduced it has become impossible to tell who is on what carrier simply by their phone number.

I noticed that http://www.jpnumber.com/mobile/ lists Docomo or Softbank Mobile for certain numbers. How reliable is that information?

Not reliable at all. I tested several numbers that I know have switched carriers and it always returned the original carrier, not the current one.

Is there some form of publicly available access to the MNP database (which should provide the ultimate answer to my question).

I don't think so. Any particular reason you need to know this info?

With most plans, there seems to be a significant difference between off-net and on-net call/text charges. So it would be nice if subscribers could know which network a specific number is on.

One way to tell what network a phone is on is by the person's cell phone email address. @docomo.ne.jp, @i.softbank.ne.jp and @softbank.ne.jp are obvious. @ezweb.ne.jp is au.

And yes, plans on virtually all major networks provide either free or discounted calling rates to same network phone numbers. In practice however, if you need to call someone then you need to call them.

If your goal is to save money, then most people will text instead of calling or use a voip app such as Line to avoid calling charges altogether.