Not really Japanese per se, but the main female vocalist is Japanese and she sometimes sings in Japanese so I allow it.
This one is from NGE: 2.0, but the version in Nichijou, and Babymetal's cover are great on their own right. I think Susan Boyle also sung an English version as well.
Speaking of Nichijou, Hyadain no Joujou Yuujou is catchy.
Bakufu Slump's RUNNER, pretty old, but it's so damn catchy.