Yes, it’s possible to walk around Tokyo. It is a very walkable city as most of its roads and pathways are paved. For example, central Tokyo is pretty explorable on foot. There are so many walking tours in Tokyo because it's really a walkable place.
If you want to walk from Shinjuku to Shibuya, it’s possible without hiccups. It might take a long time, but it’s doable. Just wear a comfortable pair of shoes, though.
Even when just walking around, you will find definitely find the most authentic and most interesting parts of Tokyo. Here are the most walkable areas in Tokyo:
Shibuya (center for youth fashion and culture)
Shinjuku (commercial center full of bars, clubs, karaoke bars, and hotels)
Ginza (shopping district with upscale boutiques and cocktail and sushi bars)
Ueno (known for ornate Shinto and Zen landmarks)
Harajuku (colorful street art and youth fashion)
Tokyo Station (near skyscrapers and historic buildings)
Ebisu (a trendy area known for Yebisu Garden Place)
Akasaka (playground for Tokyo's rich and powerful)
Ikebukuro (known for its shopping, arcades, and anime fans)
Shinagawa (famous for its ramen)
Nakameguro (famous for cherry blossoms along the Meguro River)
Daikanyama (hip area with bars and designer boutiques)
Kochijoji (known for shopping streets and standing bars)
Aoyama (one of the wealthiest neighborhoods of Tokyo)
Roppongi (upmarket entertainment and shopping district)
But of course, there are some parts of Tokyo that are not as walkable. Although Tokyo is generally safe. You need to be careful regardless. Asakusa and Ueno are nice, but they do have the highest crime rates in all of Tokyo. This is where a large number of international tourists stay. Toshima, which is where the large downtown area of Ikebukuro is located, also has a high crime rate. Don’t let it faze you. You just need to look after your belongings, especially when in public spaces.
