For full itinerary, check out my Japan Trip 2014: 9D8N Hiroshima & Kyushu Area.
Getting into Nagoya City:
1. By Train (Meitetsu Line) to Nagoya Station = limited express: 870 yen (33 mins); uSky: 1230 yen (28 mins)
2. By Airport Bus to Downtown Nagoya (Sakae/Fushimi area) = 1000 yen (about 50 mins)
We took the train to Nagoya Station. One can top up 360 yen for first class car on the limited express for extra room legs and more comfortable seats.
We bought the JR Rail Pass which allowed us to travel everywhere.
So the first thing to do at Nagoya Station => we looked out for the Japan Rail Pass Exchange Counter. We were given the JR Rail Pass passport card after filling in the required information and we had our Japan Rail Pass activated (for next day use).