I am deeply convinced that the music of Ado is the best music in the world.
Just listen to うっせぇわ. I've started listening to her after starting to learn Japanese, but now it's pretty much my main motivation to continue learning the language (through there are quite a lot).
I'm not entirely sure what you mean by alternative music (I'm not at all versed in musical styles). But in case you're looking for music made by individuals (indie music?), may I interest you in vocaloid music (music made with a voice synthesizer)? It's has quite much variety. And by snowballing (look e.g. at recommendations, other works by the producer, covers (usually by a so-called "utaite" (tried-to-sing-er)), other songs covered by a person, and various other ways) you can find more and more small-ish and small musicians. It all forms a network.
Hmm do you just mean not pop/boy/girl band type of stuff?
Shiina Ringo was pretty popular back when I was listening to a lot of Japanese stuff. She can do a lot of styles but I remember her being punk rock jazzy. In the other direction, I was also into Pizzicato Five who are more retro 60s French vibes.
I’m not sure if this fits for what you are looking for, but The Pillows are a great rock group from Japan. Some of their songs being portrayed in the anime FLCL.
The Pillows is the reason I asked for this kind of music!! They are the best!
I stumbled across their music when I was watching SoulEater. Recently I went through my old mp3-player, remembered the band and was curious about similar music, which sparked my comment.
I watched FLCL out of love for the band and, for me, this kind of anime hits just the right spot.
If I wouldn't have known about either, FLCL or The Pillows, I would have kissed your feet for this recommendation. Spot on!
I admittedly don’t watch much anime, but FLCL might be my favorite, and what got me into the band. I haven’t heard of soul eater before but I’m excited to check it out. Thanks for letting me know!
Maybe some Mongol 800 (although I guess they're more punk than alt)? They also sometimes do English lyric songs which are off enough to be funny but they still sound good.