The way I see it is that the Japanese people are extremely insular and extremely politically disinterested. This combination means that they'll usually know nothing about what's going on or how it impacts the rest of the world, so asking their opinion is a bit like warning people who don't know better about the dangers of dihydrogen monoxide.
If they hear that their prime minister is being verbally attacked, of course they're going to be shocked. And if asked if they support her, of they would! They have no tools for even considering the issues, so how else can they respond. At the same time, because of their insularity, the Japanese people aren't going to know very much about China, and most of their impressions are going to be very out of date. And so they're going to assume that any Japanese response will come with repercussions so where's the harm in any of that? Compare and contrast this with how Japanese diplomats are scrambling to even get meetings with their Chinese counterparts!
I don't think there's any chance of any war between China and Japan. The countries are far enough apart, and neither is interested in anything like that. That said, China is very eager to express their displeasure right now, so we can expect the PLA to start wearing the JSDF down with incessant patrols and penetrations of airspace. Japan is going to really hate the idea of Freedom of Navigation!
On the economic front, Japan is especially weak from all of their ongoing messes. Their one saving grace is probably the auto industry, but this is also the industry most vulnerable to Chinese actions. For China, this is basically a godsend because they're given free reign to bully an American ally who has decided this is a great time to have no friends.
