Are they very sensitive about this topic?
I am not the person to give an objective answer, I don't live in japan and I don't even speak Japanese. But from what I can see (from afar) is that there is a certain amount of nationalistic movements as there has always been. So no change there - there will always be people who are not happy with foreigners.
But in real life there are more foreigners who come to live to Japan, especially from asian countries as their physical appearance makes it easier to blend in.
I was surprised to know that Renho Murata who became the leader of Japanese opposition Democratic Party and who may become the first female Prime Minister became Japanese only at 17 years old. She was born in Japan but her father was a Taiwanese. And she seems to be all about diversity and coexistence.
So I would say that Japan is gradually changing and the Japanese are rather ok with that since they support such leaders.
In many respects, LDP the leading japanese political party, could be labelled as "Far-right". Japanese emigration laws are one of the must severe in the world, so, they don't really need "anti-immigrant" movements as we know it in Europe. As the policy is changing now we'll see what they reallyfeel
I think they do, actually. Not only "nationalists" or the old school yakuza, but the average people are not get used to the quantity of Chinese tourists in the last 2-3 years. They're not aggressive normally, but they fell rather perplexed. So, I think the conflict between traditional japanese "insular" way of thinking and acting and the government policy to make Japan "more open to the world" will only grow.
There are some people who are unhappy about immigrants, and also about Chinese citizens buying large amounts of land in Hokkaido. But this is a really small group of people.
The problem is that there are a few politicians who rely on the block vote of these people, and the nationalists. So, they occasionally go on air saying something anti-immigrant. But the mainstream media usually jump on them and get them to apologise.
Last year, a group of people hosted a "Japanese Only" banner at one of the league soccer games. The home team apologised, and played their next game without spectators, to compensate.
Over the twelve years I stayed here, I have noted that the demonstrations - and trucks with loud songs - by the nationalists have been on the decrease. Most anti-immigrant demos you see in Ueno are paid performances by the gangsters.
Overall, there is nobody who call immigrants "Foreign Trash" (like they do in Singapore). There are some cases where foreigners cannot rent apartments or enter clubs, but these are on the decrease.