I can give some personal experience on what you'll see with tattoos in Japan, but as for why they would ask, my best guess would be for placement. In my experience tattoos are fine in cities (at least the more westernized cities that I've been to, since I've only been to Tokyo, Kyoto, and Osaka, and it was winter in Kyoto and Osaka so I only had my tattoos visible at night clubs).
Tattoos definitely have some negative associations, as Cliodhna mentioned they traditionally are related to Yakuza. You'll still see a fair number though in some cities. I saw several in the few times I've been to Tokyo. If you have some you'll probably get some stares, but it probably won't be much more than you would get for any other interesting feature (really blond hair or red hair, really tall, etc). I have a few small tattoos on my forearm and a middle sized one on my shoulder and at least in Tokyo I never had a problem. I saw one guy kind of staring on the train and the capsule hotel I stayed at technically had a no tattoos policy in their bath, but I was told it was okay if I covered it and after seeing a Japanese man with an uncovered tattoo in the bath I just didn't bother and no one cared.
I've also spoken with multiple younger Japanese people who are in the US (19-21 year old exchange students and 30-35 year old women who are here because their husbands have jobs in the US) and it seems the view of tattoos are changing a bit. Admittedly they seem to be from more urban areas (Kyoto, Tokyo, Nagoya, Osaka) but in their opinion, western style tattoos are essentially a non-issue to everyone they know, maybe even are considered stylish. You'll almost certainly have to cover them at some places like Onsen, public baths, and gyms, and some will probably refuse you service, but I don't imagine you need to expect a big scene.
I would absolutely advise making sure you have a way to cover them. Since I have a set of 5 small tattoos on my forearm I fully plan to wear long sleeved shirts year round in school if I get a position unless I am explicitly told by my CO that they don't mind if I show the tattoos without me having to ask. I recommend the same to others, just because it would be the polite thing to do. I saw a post here two years ago (I think from a coordinator but it's been a while) when I applied the first time saying they do not discriminate in hiring based on whether you have a tattoo and I wouldn't personally say you should get it removed. If you expect to stay in Japan a long time you might eventually consider it, but from what I've seen the cremes don't fully remove them (they leave them very faint but visible) and lasers are expensive, painful, and take multiple sittings.
Good luck to those who applied, and if I make it in and so do you I'd love to see the work that the other people with tattoos have, and I'd love to keep in touch about the way that they influence our experiences.
