I will be spending four weeks in Japan, visiting all of the "main islands" except for Hokkaido. Some of the activities would be hiking across the woods and mountains as well.
The vaccine is for "Japanese Brain Fever" or Japanese Encephalitis.
The short answer is no, you don't need it.
The long answer is that all Japanese have been vaccinated for it since the 1960s and statistical incidence is down to background noise levels: this study figures less than one case per year since the 1990s. So unless you're planning on spending a lot of time in rural pig farms, it's not worth worrying about.
The CDC is, as ever, more cautious, but even they state that "vaccine not routinely recommended for travel limited to Tokyo or other major cities". The Australian Dep't of Health goes a bit further, stating that "the risk of travellers in Asia acquiring JE is extremely low" and only recommends vaccination for "travellers spending 1 month or more in rural areas of Asia" (unless there are significant other risk factors).
All that said, the vaccine is reasonably effective and provides lifetime protection. So if money and needles aren't a concern and (probably more importantly) you plan on traveling in places where the disease is more common, like Vietnam or Indonesia, you might as well go for it.
The French governmental website "advices to travellers" says about Japan (my translation):
Just thought I'd give a U.K. perspective.
You almost certainly don't need the vaccine, particularly if you're visiting soon - the transmission season is June - September, except in Okinawa / Ryukyu islands where it's April - December.
I was advised by the gap year charity I went to Japan with to get these vaccines, but my doctor informed me that they were unnecessary (for Tokyo).
More specifically from the U.K. Government on who is at risk:
and on risk management procedures
From The National Travel Health Network and Centre, part of the U.K Health Protection Agency.