You can't do Tokyo in 24 hours, you are going to miss many of the "do-not-miss" anyways. So you will have to make choices.
First thing to realize is that Tokyo is not really a single city but a prefecture, it is like a bunch of smaller cities (wards) stuck together, and there are 23 of them. They all more or less independent and they have their own center, and they also have some kind of theme. All that to say that there is no "downtown Tokyo", and choosing what to do in Tokyo is essentially about the districts you will choose to visit.
Which one depends on your interests. Shinjuku which is a major business center at day and a shady (but still safe!) place at night, Shibuya for fashion and youth, Roppongi for foreigners and its nightlife, Asakusa for its temple, Yoyogi's park, Akihabara "Otaku" mekka, Odaiba's futuristic vibe, Ikebukuro's shopping centers and arcades, Ginza's luxury, etc... Tokyo-Edo museum is indeed awesome if you are interesting in the history of the city, and it certainly a "do-not-miss", but on a 24-hours stay, you have to make choices.
If I had to pick one to add to any list, maybe Shibuya crossing. It is one of the most famous sights of Tokyo and a symbol of the busy city it is. It is also right out of the station on the Yamanote line, so close to other notable places. Just north is Harajuku, Yoyogi, and Shinjuku, all worthwhile.
Far from an expert, but two favorites from last time I was there:
- Uneo Park and the Tokyo National Museum.
- Tokyo Metropolitan Edo-Tokyo Museum (and I've heard great things about the open-air take on this museum in the suburbs ... which could be a better option since the former I just saw is being renovated).
And on the other side of the country, don't miss Kyoto. Just came back from there, and I'm not sure I've ever fallen in love with a city faster.
But above all, I adore just walking around any city, aimlessly, and exploring whatever I stumble upon. Turn into a temple, walk through a random neighborhood grocery store, delight in the ordinary-for-here that's not-so-ordinary for elsewhere.
If you're visiting Shinjuku, nearby there's a narrow street called Omoide Yokocho. Just take in the vibe and choose a yakitori spot to grab a bite and drink your poison of choice (tea/beer/sake). I would recommend going at night/dinner time.
Speaking of Shinjuku and videogames, if you've ever played any yakuza/like a dragon game, you owe it to yourself to go to Kabukicho and its big red gate.
In any case, whatever you choose to visit in Tokyo, it will be really nice, and a lot of it will still be waiting when you eventually come back.
Check the industry in Kawasaki or maybe Minato Mirai (same direction). Also Enoshima makes a nice day trip from Tokyo (don't miss Shonan Monorail and Enoden on your way there). :)
