With my African textured hair and knowing the stereotypical East Asian hair, the FIRST thing I'd think of was my hair! Unlike much of Europe, Japan has few Black people and even few people of color who are not also Asian.
As I'm natural but I blowdry, I'd take my attachable blow comb. I use a Willie Morrow and those things are hard to get almost anywhere, so w/ Japan I'd take 4-5 even just to store tho I know how primo space is in Japan. After that, my Solano dryer, tho I'm sure you can find an excellent dryer in Japan. This obviously applies to flat irons, but if you use a hot comb...pack it and buy the necessary adapters. Then find places online that can sell a replacement in a pinch.
Of course, there'd be my conditioners...I'm not so fussy about shampoos tho I have a few US brands I like (Motions namely) and you can get many French products in Japan and I like their shampoos. From the US, I'd def take Mizani, Goldwell, and my Aphogee protein treatments. If I could squeeze it in, Motions products. I rotate all of these as needed. If I wanted something more old fashioned like Suplhur 8, Glovers, and the like...I'd buy some and arrange for family to send at regular intervals.
If you are relaxed, you'd obviously want to take your relaxer of choice and try to get as much w/ you as possible. Relaxers will be the hardest thing to get in Japan and what's there will be terrible for your hair unless it's vastly overpriced cuz it's so rare. If you prefer stuff that's fairly easy to get in the US like Dominican products or Ayurveda...you might take what you can carry w/ you, but have a nice supply already boxed up to be sent to wherever you end up (family or friends needed here). Then you'd find good online sources as your plan B. Dominican will be almost impossible to find in Japan, methinks...you have a better shot w/ Veda.
Of course, you can start to explore some local products (never relaxers, I'd say)....Japan has 1st rate products if you can find what fits your needs. Camellia oil is great for hair and cheaper there than here. If you use certain oils (says coconut) or butters (shea/avocado/etc) , take them and then make arrangements for replacements. Find out what oils are easily available in your world...the Japanese don't normally use much oil in their hair, so premade stuff or multi varieties of natural oils may be hard to get unless it's stuff like Aveda. I'd think olive is pretty easy to get, but peanut is fine too.
Whatever you do, stick w/ things that are fairly basic in the US and someone can easily buy for you. Before I went overseas, I'd experiment w/ cheap(ish) US products that are easy to get. Your family and friends may have trouble w/ some of the more obscure or net based stuff.
Even tho it'll cost you more and you have wait times, you can get most of what you need online these days, so no real worries. Family/friends/connections are the best option and cheaper tho. Maybe you can think about making a connection here...find a member and arrange to get stuff sent to you between yourselves.
As for skin care, I'd take NOTHING. Japan has the best skin care...both natural and store...in the world. They have every kind of product you want including fade cream for blemishes.
(Sorry for length, but your question is complicated seeing as how you are considering living there.)