ASAKUSA! If you want to experience what historical Japan is like in Tokyo then Asakusa would be best place to visit. You can visit the old reconstructed Sensoji Temple and walk around in the Nakamise Shopping Street where there are many food and souvenir stands selling traditional Japanese stuff such as yukata, fans, artwork, pouches, charms and etc. I would recommend going to Asakusa and then walk to the Tokyo Skytree tower which is literally about 15-20 minutes walk away.
Kaminari Gate.
At Asakusa you can find all kinds of really cute traditional Japanese goods such as charms at an inexpensive price. Great place to shop for souvenirs for family and friends!
There is also an endless amount of local Japanese sweets and snacks that you can find here at a really affordable price. So many cute and unique snacks to choose from. I wanted to buy all of them!