Get fruit at AmeYoko
Not just for fruit but one of the most culturally interesting and inexpensive places to shop in Tokyo is AmeYoko market located right by JR Okachimachi Station, as well as accessible by Ginza (Ueno Hirokoji station), Oedo and Hibiya lines (at Naka Okachimachi station).
Fruit, seafood, clothing, nuts, candy, watches, cosmetics, Japanese military clothing.... you name it and it can most likely be found at AmeYoko. Pretty much any shop or stall on the long stretch of market street is open for negotiating prices, a rarity nowadays in Japan.
AmeYoko derives its name from the Japanese word for candy (ame) and also the first two syllables in "America". It was a shopping district mainly selling candy that came alive during the US postWWII occupation and diversified its offerings.
You can get many types of fresh fruits and dried fruits at much better prices than any supermarket chains.
Also as a general rule of thumb, to avoid high fruit prices avoid shopping at upscale supermarkets (e.g. National Azabu, Kinokuniya or any premium dept stores) and buy at little mom & pop green grocers further from high profile stations. Or go to AmeYoko, Niku Hanamasa chain stores or even Costco if you need large quantities at reasonable prices.