It all depends on the employer, and how honest you are about it. I have a few tattoos and I have not faced many problems because of it. But I can cover mine up by wearing a 3/4 sleeve, if you can cover yours you have no problem. I cover them while working, as I would have back home, but my employer knows they are there.
Working as an ALT can be problematic however, since they might replace you should they find out that you are tattooed. I've hear of one case where the ALT was spotted in his/her freetime by a student, word got around and job finished. I'd say the piercings have to go however.
Many people seem to think that you are crippled in Japan if you have tattoos, that is not my experience at all. There will be some troubles, sure, but mostly it won't affect you. A lot of japanese young people have tattos too, if you look for it you see them everywhere.
My biggest problem is that I am not allowed to pools.
Def in kind of right about the 50's though I guess.
Working as a teacher in Japan you are expected to look nice. No beard, no extravagant hair styles or colors, clean but discreet make-up, no shorts, no "weird" clothes, no bare feet and so on. In such an environment showing off tattos and/or piercings is obviously not going to land you a teaching job. I suppose the same goes for an office wi an even stricter dress code added.
Frank is also right about there being other hurdles too, and more important ones...