Ah, joejones, shouldn't we be doing practice MBE questions?
Another non-secret (unless you're not paying a lot of attention) is the Ramen Museum - which is also a great place to take anyone who enjoys eating. You might not think of it as a secret, but it's a great place to show off to folks, especially when they're only in Japan for a short time. I took some friends here from Tokyo who were passing through on their way to China (they stayed for a day) and wanted to try a quick trip on the shinkansen (~$15), eat, shop, and then be back in Tokyo by nightfall.