For traveling from Shinjuku to Shirakawago via Matsumoto and Takayama, using the 'Three-Star Alps Unlimited Shinjuku Ride Pass' is the best way to go.
Let me answer to your questions one by one.
1. The difference between type S1 and S2 is whether or not you can go from Shinjuku to Takayama directly instead of changing a bus at Matsumoto to go to Takayama. I would recommend that you buy type S1 because you can go straight to Takayama from Shinjuku without switching to a local bus which runs only 6 times a day. Also, you can purchase type S1 at the Shinjuku Expressway Bus Terminal where the bus departs. On the other hand, for type S2, you have to walk down to the different ticket office. It should be easier for you to get everything at one place.
2. You need to book your ticket from Shinjuku to Takayama and from Matsumoto to Shinjuku online in advance from the following link.
When you purchase the ticket at the ticket counter, please mention that you want to buy 'Three-Star Alps Unlimited Shinjuku Ride Pass Type S1' .
You will receive tickets from Shinjuku to Takayama and from Matsumoto to Shinjuku and a voucher for the unlimited free-pass. When you arrive at Takayama bus terminal, please show your voucher and pick up your free-pass there.
You do not need to make a reservation for the local bus between Takayama and Shirakawago and between Takayama and Matsumoto.
I hope this is clear enough for you.
Enjoy your trip in Japan and let us know how you liked it when you come back.
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