Hi Shu,
If you do the travel you have listed, I think I'd aim for a 7 day rail pass for use from the 4th-10th.
There are 4 sleeper trains that you should investigate:
- Cassiopea &Hokutosei (not 100% free)
- Akebono & Hamanasu (possible to be free for JRP holders).
Cassiopea and Hokutosei will take you directly to Hokkaido from Tokyo, but will require you to pay for a couchette or room (JR Pass will cover most of the base train fare) as a minimum.
Akebono has goron-to-shito couchettes that are free for Japan Rail Pass holders to use, but will only take you as far as Aomori, where you will have to change train to go into Hokkaido. Hamanasu is a sleeper service that runs from Aomori to Sapporo that has carpeted areas (nobi-nobi seat) free for use for Japan Rail Pass holders.
A common trip I have heard Japan Rail Pass users do is to use the Hayabusa/Hayate shinkansen up to Aomori in the morning and then hop on the Hamanasu in the evening for the trip up into Sapporo.
Please remember that it is very important to reserve your place on the Sleeper trains, so try to reserve your place as soon as you arrive in Japan and exchange your Rail Pass.
Based on this method of transport, you itinerary from the 5th would look something like:
- Tokyo > Shin-aomori: ¥16370, (203mins, 713km)
- Shin-aomori > Sapporo(jr): ¥9000, (445mins, 483km)
- Sapporo(jr) > Shin-aomori: ¥13200, (323mins, 483km)
- Shin-aomori > Tokyo: ¥16370, (206mins, 713km)
- Tokyo > Osaka: ¥13750, (194mins, 556km)
- Osaka > Tokyo: ¥13750, (194mins, 556km)
TOTAL: ¥82,440 for 6 legs
At ¥28,300, a 7 day rail pass will give you a huge amount of saving of ¥54,140 over single tickets. Please remember to check the timings of your trip as some of these legs are long distance and so this will affect your planning to some extent.
Hope this helps!