Hi,
I was kind of in the same boat in Tokyo. Had lots to do/see, but not as much time as I liked. If you're willing to go from early morning to evening you could based a day on the route I did:
7AM: Tsukiji Fish market for breakfast: rather touristy and be prepared to wait an hour on line for a seafood breakfast, but definitely a different experience to have (btw, if you want to see the tuna auction, arrive by 5AM)
AM: From Tsukiji you can walk through Ginza towards Tokyo station area and the East Imperial gardens: Ginza is obviously closed early morning but you can wander down the small gallery streets and the broad department store avenues. Maybe you'll catch the Baum Kuchen rotating and baking in the window...
AM: Visit the Imperial gardens: It's a good place to rest; not a lot to see, but nice green spaces and it's free.
AM-noon: Walk from the Imperial gardens to controversial Yasukuni shrine (which is quite nice actually)
Early afternoon: Train to Asakusa. Here's the famous Kaminarimon gate and Sensoji shrine. Plus Nakamise street has tons of souvenir/'traditional' craft stalls and a few streets west, Kappabashi, has tons of restaurant gear, including ceramincs and delicious looking fake plastic food.
Here you could think about visiting the national musuem at Ueno park, especially if the rain is getting to you.
Late Afternoon: Train to Harajuku (I'd skip Shibuya, which is filled with department stores, with admitted cool gear). But Harajuku is more interesting for me; it's where you can watch 'hip' teenagers strut about and pack Takeshita dori street looking for weird accessories and Engrish T shirts.
Walk from Harajuku to Yoyogi park, which may be more unintereseting in the rain, in which case head over to Meiji shrine.
Late afternoone-Evening; Either walk or train up to Shinjuku, where you'll be crushed by mobs of commuters and lost among the different sections of Shinjuku station. A walk around that area including peeking into the Golden Gai (rows of small, small bars serving regulars) and then wandering through a few streets of Kabuki-cho (despite being a red-light district, it's really lively and totally worht a gander.)
Evening: After sunset you can swing by the Metropolitan Office Building for a free view of the city from their 60th floor observatory. It's a surprisingly nice view, but only 1 of the windows really faces Tokyo tower, etc, as the others are taken up by a restaurant.
Alt. Evening: Depending on where you're staying, I really enjoyed wandering Ameyoko arcade in Ueno. It's got cheap clothes stalls, yakitori booths, takoyaki stands, ramen stands, fruit and seafood vendors. It's a bit more down to Earth, compared to all the glitz of higher end shopping streets.
And finally, you can party the night away around Tokyo. I'm not a clubber, but many foreigners head to Roppongi, though I think Shibuya looked more interesting for a more immersive experience.
Sorry, it's so long; I keep blabbing on, but have fun whatever you do.
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