You can get reserved tickets for any carriage - the green car is like first class. When I traveled to Kyoto we reserved our seats in a non-smoking carriage. The problems come on national holidays if you don't have a reserved seat - the trains get jam packed - not pleasant if you're traveling long distance.
Kitkatgo is right - I would always reserve my seat - it's just not worth the hassle.
There's a great site here for more info on the
bullet train