Here is an itinerary I suggest.
Day 1 Arrival at Tokyo
Day 2 Nikko or Kamakura
Day 3 Mt. Fuji, Hakone
Day 4 Shirakawa-go, Takayama
Day 5 Tokyo City Tour
I have no intention to change your itinerary, but if I were you, I would make my itinerary like that. (I am usually a budget traveller, having short vacation time.)
Day 2: Most popular tourist attractions near Tokyo are Nikko, Kamakura, and Hakone. Pick out Nikko or Hakkone for the second day.
Tourist Attractions around Tokyo:
http://japan-guide.com/e/e2164.htmlDay 3: Leave Shinjuku for Kanazawa by night bus.
Day 4: Leave Shirakawa-go at 1400. Leave Takayama for Nagoya on the bus at 1730. Even if you take the bus at 1900, you may make it to a night bus from Nagoya to Tokyo, but you must take delay into consideration. Leave Nagoya for Tokyo by night bus.
Day 5: It may be better to stay and make a tour of Tokyo, because you are departing from Haneda on the fifth day.
My itinerary may look hard. Please reconsider an itinerary for yourself which is suitable for your time, budget and fitness level.
P.S. It is an idea to go to Hakone on the second day and leave for Kanazawa at night. Kanazwa also has some good tourist attractions for your third day. Leave for Shirakawago in the afternoon. Stay in Shirakawa-go overnight. I believe it must be a great experience for you. You can spare the fourth day for Takayama.
Kanazawa:
http://japan-guide.com/e/e2167.html