Hokkaido certainly has more open space than anywhere else in Japan, but it's not the only place to find great scenery and open roads. Much of Tohoku, the Japan Sea coast, Shikoku and Kyushu are rural and drivable.

I think the most "wow" drive for me was around Shiretoko and Akan national parks (in Hokkaido). But I also think that central Honshu (say Yamanashi-Nagano or Gifu-Ishikawa) are great places to get off the beaten track and explore.

For driving immediately after landing at NRT, I think that you could get into some interesting country quite quickly if you head for the coast and then turn left (North).

A lot of people told me they enjoyed watching the "race across japan" by TopGear on YouTube. I guess that might give the OP some idea of what to expect (although it's probably rather contrived ... I ought to watch it myself one day).

To make the advice in this thread a little easier to interpret, I think people who've never actually driven in Japan should say so at the start of their posts. Advice such as taking a "shinkensen" to Kyoto and admonishments that STRONGLY RECOMMEND NOT renting a car ... I think it would help if we knew if such advice is based on actual experience ...