I studied abroad at Waseda University (also in Tokyo) in 2005.
The school did productions of Little Shop of Horrors, Urinetown, and Avenue Q all on one weekend. Avenue Q was still running on Broadway, and Urinetown was likely new to being licensed.
Legality aside, I had a really fun time. I couldn't see many shows as a college student, and was really happy to see them staged. I remember the Avenue Q cast rolled out a banner with a translation for the phrase "It sucks to be me." The shows were done fully in Japanese, but that phrase was said in English.
This was the early 2000s, so it was much easier to fly under the radar. Japan also has a culture of a grey line around copyright and fan works (for example, fan artists frequently sell art and manga of popular anime, video games, etc).