Since you will have already done a 12 day cruise by then, I've listed some less touristy activities:
How about a free local guide as a chance to meet one-on-one with a Japanese: http://www.tokyofreeguide.org/
Take a taiko drum lesson: http://www.taiko-center.co.jp/english/class/tokyo-class.html
For the best ever tempura at Tenichi: http://www.tenichi.co.jp/mainshop/index.html
It is expensive, best to go for lunch. Ask for a seat at the counter, the cook will place your food right from the fryer to your plate. How fresh is that?!
Experience high quality sushi at Ginza Kyubei. Best bargain is lunch. They take reservation only for the 11:30 sitting. But you could wait for the second seating.
3 hour architecture walking tour with Context Travel: https://www.contexttravel.com/cities/tokyo/tours/aesthetics-stylizing-modern-japan The tour ends at Roppongi. If you take the morning tour, then you can stay and explore the area.
Museums: Edo-Tokyo Museum, Tokyo National Museum
If you want kitchenware go to Kappabashi, an entire street of shops: http://www.japan-guide.com/e/e3020.html
Askakusa is nearby, if you still need a shrine fix. You can combine these into one trip. There are food stalls near the shrine.
You could continue on across the river to Tokyo Sky Tree: http://lang.tokyo-skytree.jp/en/
Walk along the Ginza on Sunday when it is traffic-free. Go to the basement of any major department store and see the amazing foods and fruits.
Go to Mitsukoshi Dept Store when it first opens. The staff will bow to you as you walk down the aisle. The closest you get to feeling royalty.
Go to Yurakucho for yakitori stands.
Go to Tokyo Station and try ramen noodles from various parts of Japan at "Ramen Street". This may be hard to find since the station is so big. Go in the central entrance with the Daimaru Dept Store on your right as you face the station. Ramen Street should be on your left after you enter. But ask someone if you can't find it.
If you like eel, ask your hotel to recommend an eel specialty restaurant.
If I were to return to a cat café, I would take the opportunity to interact more with the Japanese there instead of the cats.
Here's a preview for you about Tsukiji: https://www.youtube.com/watch?v=tuOkrOcZ6nQ&index=7&list=PLFuc_7VVFh6oHm5i3berXoZdkFcjZSYYm
I think that's enough suggestions to occupy you for 4 days. Have fun. I envy you. Would love to return to Japan.