DD, if you want “new” Tokyo, on a weekend, simply hang out around Shibuya and Harajuku. Very trendy, lots of cosplay, and in Harajuku you can retreat into the Meiji Jingu shrine. The best food for the money is “picnic” style, buy okazu (side dishes, since rice is THE dish) in the B1 food courts of department stores, and carry it out to parks, or since you’re gaijin, sit in the outdoor area (WX permitting) of a coffee shop where you can order a cheap NT150 latte (Y500), and nibble from your cache.
If you want old Japan, take the train from Asakusa ( not Akasaka) to Nikko, the mausoleum town of the Tokugawas. Temples, ambience. Sat. would be better than Sun., slightly less crowded. Visit Asakusa temple before or after the train trip.
For that special meal, make it Japanese, since all non-JP food is warped in Japan. Rather than try to name a restaurant that requires navigation, cruise the food court floors of the largest depatos (department stores) in Shinjuku and Shibuya. There are all ranges of prices and views.
For curiosity, find an “old style” coffee shop that uses the Chinese characters for KA FEI, and it won’t have espresso, but some interesting regulars, ambience, Beatles on the sound system, and the alchemical approach to drip coffee using… gasp … foreign beans. About 6 p.m. these old style coffee shops typically turn into soju parlors. The counterhelp even change clothes to reflect the transformation. There will be some regulars who speak English who will become chatty during the soju phase. BTW, these old style coffee shops are one of the best venues for language exchange in Nippon. Classier than a bar, no airheads, almost a salon.