The cheapest way to get to Nagoya from Tokyo is by plane, that is if you are willing to ride low-budget airlines and are able to score a good deal on plane fare. Try checking Jetstar, Vanilla Air, or Peach and see if they have ongoing promos. If you are lucky enough, a round-trip plane ticket may cost you just under 10,000 yen per person.
But if there are no deals for your desired travel dates, the next cheapest option is to take an overnight bus. From what I know, one-way buses cost around 3,000 yen per person going to Nagoya. The only downside of this is the long hours spent on an uncomfortable bus seat.
If all else fails, take the shinkansen. But for around 15,000 yen per person just one-way? MEH.
As for accommodations, I’d highly suggest mid-class Japanese ryokans. You can also opt for the hotel near Nagashima Spaland if you are planning to visit that area too. If you’re going this winter, don’t miss the Nabana no Sato illumination!
Hope you enjoy your travels!