It really depends on how deep your pockets are. For someone affluent, this is a very easy trip on the shinkansen bullet train. For those on a budget, Nagoya will seem that much further. 'Far away' is relative when it only takes 1 hour and 40 minutes for those willing to spend approx 10,000yen each way and 8 hours for those who can only afford the 3,000yen bus fare (there are in-between and mix and match options).

If you've got a visitor's visa, you could always factor in a day in Nagoya as part of a week long trip around Japan with the JR pass, this could be quite economical, particularly if you're hoping to make that visit to Nagasaki or Hokkaido.