If you like music (and science tells me that you do), then you'll like listening to music in your target language. And if you like doing something in your target language, do it.
- Find music in your target language. To start with the obvious, you'll need to first find some music in your target language. You can typically find some with simple Google searches or by asking a native speaker for some recommendations. Once you know what you're going to get, getting it can be a bit more challenging but pretty much any artist has videos up on YouTube and music that can be downloaded from somewhere on the internet.
You'll find more details on how to do all of the above on this post on Yearlyglot (including the comments!) and apparently Spotify is pretty useful as well. And for a great explanation of how to get Japanese music, check out this post on Tofugu.
- Figure out what songs you like. Once you've found some music, it's time to figure out which are the songs you like the best. Why? Because these are the songs you can put on a favorites list and listen to over and over again—thus getting lots of exposure to all the vocab, grammar, etc., used in the song. I keep track of which songs I like the best by using iTunes' star-rating system; any songs that I really like get a 5-star rating.
- Make a playlist that includes just those songs. This can be done manually, but I'd recommend using a smart playlist—make it once and you're done forever. If you use the star ratings and you add more 5-star songs or get sick of a 5-star song and downgrade it, the list will automatically update to respectively include or exclude the new or downgraded songs.
Making rules to find all of your songs in a given language can be a challenge. My system has been simply adding the name of the language to the front of the genre field, so instead of "Rock" it'd be "Japanese Rock", etc. I've been doing this ever since I first got iTunes way back in 2001, so pretty much all 9,000 or so tracks in my library have this, which makes it simple for me to find songs of a particular language. The downside of this is that you have to insert the language in the genre field whenever you import new music.
Another method is to simply dump all your target language songs into a simple playlist. To get your favorites playlist, you'd then just make a rule containing anything in that playlist with a 5-star rating. The downside of this one is that you'll have to make sure you add any new target language music to your target language playlist.
Another alternative is to use the "Grouping" field. No one quite seems to be quite sure what that field is supposed to be used for, but people use it in a variety of ways. You could use it to add the language of the song, and then use that to create smart playlists. This would function like my genre method, with the same downside; you'll have to add it in for each song.
- Find the text of the lyrics to the songs you like, which is pretty easy to do generally, although Japanese presents some unique challenges. Unless your music collection is of a very manageable size and contains few albums with B-side tracks that you're not that fond of listening to, I wouldn't recommend trying to get the lyrics for all your songs at the outset. You only want to bother tracking down lyrics for those that you know you'll be getting the most out of, i.e., your favorite songs that you'll be listening to again and again.
- Look up the words you don't know in the lyrics using Learning with Texts, import them into Anki, and learn the words in Anki. Listening to music loses a lot of its value if the exposure isn't meaningful, i.e., if you don't know the meaning of what's being sung. The purpose of this step is for you to learn the words you don't know by first looking them up (using LWT) and then memorizing them (using Anki). Using LWT and Anki together for this isn't the only way to go about it, but there's no way I know of that'll do it for you faster.
- Copy and paste the lyrics into the "Lyrics" tab of the song's info window in iTunes.
Not only will this quick copy-and-paste allow you to pop up the lyrics anytime you're listening to the song on your computer, but, if you've got an iPhone, the lyrics can also be displayed on your iPhone's screen whenever you play the song on your iPhone.
- When you're listening to the song, read the lyrics. You'll be able to do this whenever you're on your computer or on an iPhone.