Hi there,
Nagoya is right between Tokyo and Osaka, however I think that your current itinerary order is a good one. Will you be flying out of Nagoya as well?
I think that the 7 Day JR Pass would offer the best value, if you return to Nagoya. The problem is that it would only pay off if you can visit all of the above places in it's 7 days of use. Otherwise it may be better to buy normal tickets if you wish to take more time to travel around.
You could do the following itinerary with a 7 Day JR Pass.
April 9 - Arrival Nagoya
April 10 - Explore Nagoya, you can for instance visit the castle and the JR train musuem.
April 11 - Activate JR Pass, head to Tokyo.
April 12 - Tokyo
April 13 - Tokyo
April 14 - Day visit to Hakone and late train to Osaka
April 15 - Day visit to either Kyoto or Nara
April 16 - See Osaka or Kyoto
April 17 - Stay the day in Osaka and take a late train to Nagoya (last day of the JR Pass)
April 18 - Last minute shopping and flight back home.
You can use the JR Pass between all cities and use the Shinkansen (bullet train) for going everywhere (only Nara is reached by local train).
I hope this gives you some ideas!
Daniel