As elsewhere in the world, bottled water has become very popular in Japan. While it is surely better for you than the other choices which amount to nothing more than sugar water, I still have trouble paying so much (usually about 120 yen, 500 ml PET bottle) for something that comes out of the tap for next to nothing.
Here is an interesting offering. The Tokyo Metropolitan Government Bureau of Waterworks is selling bottled tap water at a modest discount. Most 500 ml drinks in PET bottles sell for either 140 or 150 yen. Bottled water sells for around 120 yen. I found this Tokyo bottled tap water on sale in Tokyo Station for 100 yen.
From the label, you can see that they are trying to make a point about the purity of Tokyo tap water - "世界に誇る東京水" and "超高度浄水の水道水". Roughly, "World renowned Tokyo water" and "Ultrahighly pure Tokyo tap water".
Also, in English on the lower edge of the label is written "TOKYO WATER is purified with the advanced water treatment at the Akasa Purification Plant and is bottled without chlorine."
100 ml: sodium 1.87 mg, calcium 2.13 mg, potassium 0.23 mg, magnesium 0.42 mg
100 yen for water is still too much, and the mountain of PET bottles that results is a problem, but it is still better than the other bottled water options.