Home to the Chunichi Dragons, the Nagoya Dome is located in away from the main area of Nagoya. The ballpark is surrounded by a parking lot and various industrial and office buildings. if you are going to the game, just plan on going to the game and then exploring other areas of Nagoya during your visit.
The Nagoya Dome is a fairly sterile circular dome stadium. There is not much of interest within the seating bowl. You can take a walk to the second floor (technically labeled 3F) and find a food court with Japanese style restaurants and a small Dragons museum which features many of the trophies and pennants from their championship years.
The easiest way to get to the game is to take the subway. The purple Meijo Line will drop you right off at the Nagoya Dome mae stop. Follow the crowds out Exit Number 1 along "Dragons Road." You will see lots of photos of the team on the walls of the tunnel leading to the Nagoya Dome. Once you get outside, the dome will be right in front of you.
Nagoya Dome Official Site (Japanese)
Dragons Website (Japanese)
Wikipedia Info (English)