Japan Radio lets you pick from a selection of Japanese music radio stations and listen to them in the background on your device. The name of the song is shown on screen whilst it plays. You can also record sections from whichever station you're listening to and play it back again later.
Other features include a chat section and relevant videos that were recently put on YouTube.
It might not be as polished as Spotify, but this isn't half bad for a free music app!
Apple store page: https://itunes.apple.com/en/app/japan-radio/id418701864?mt=8