Whilst many people are familiar with the driving ranges at Lotte Kasai or Jingu Golf Club, some of the smaller ranges often go unnoticed on the roof-tops of buildings throughout Tokyo. In Shibuya there is a range about 5 minutes walk from the station in Nanpeidai.
The website for this range is at http://www.nanpeidai.co.jp
The range is not large as you can see from this photo but if convenience is what you are looking for this is just a short walk from the station and easily accessible. Lessons are available also at the range.
For an English list of many driving ranges throughout Japan you can also try this website.
Follow up to this post, it seems the building has been knocked down and the driving range no longer exists. I will be on the look out for a new range somewhere in Shibuya.