I recommend taking a highway bus from Shinjuku because you can enjoy much better scenery at a low price.
As to Shinkansen, You can take a Tokaido Shinkansen from Shinagawa sta. to Nagoya and take a JR Takayama line from there. The other way is to take a Hokuriki Shinkansen from Tokyo sta. to Toyama and take a JR Takayama line from there.
*You cannot take a Tokaido Shinkansen ‘Nozomi’ with the Japan Rail Pass.
In terms of time, Shinkansen & train is about half an hour to one hour faster than highway bus. However, the scenery is absolutely much better on a bus.
You can view Mt. Fuji, Mt. Yatsugatake, Lake Suwako, and Central Alps from a bus, whereas you can only see industrial scenery in Tokaido Shinkansen or you are in a tunnel most of the time if you take a Hokuriku Shinkansen.
I hope this help you make a decision for your transportation.