Back in the 80's when I took Korean Air between Tokyo and New York City, I used to get a box of kimchee and mentaiko (pollack roe) at Kimpo Airport on my way back to Tokyo. Even those airport souvenier shop kimchee was much better than those you could get at a local supermarket in Setagaya.
And in the 90's, I worked in Azabu Juban for awhile. And there's a Korean grocery store right by the Korean Embassy. They had really good kimchee! I think there are quite a few authentic kimchees available in Ame Yoko market in Ueno, too.
In Osaka, there are quite a few Korean populations in the area, and I'm sure you can find good ones there too.
In my LA/San Fernando Valley neighborhood, there's a huge Korean market where I go for my kimchee shopping. I usually get a small jar, but you can get a huge bucket full of kimchee, too! There are several varieties of kimchees, in addition to cabbage variety. I have no idea what most things are, but fun to look!
Kimchee has been popular for many many years, but those Korean TV dramas may be fueling the interest in Korean foods in Japan.
Japanese are good at manipulating foreign dishes. In other words, sometimes it's hard to find authentic restaurants. It's considered good that the recipe is changed to match Japanese palate. I personally like to try what the natives eat, though.