Hey there!
The sakura shinkansen will stop at Kumamoto so you will be able to get off there. From Kumamoto you should aim to get the Kyushu Odan Express bound for Beppu on the Hohi line: http://goo.gl/wYeXx. The Aso station is about an hour by train. Your Japan Rail Pass will certainly cover getting you to JR Aso Station.
Tourist Information
Once you get to JR Aso, it's really worth heading straight out of the station, turning left, and making a bee line for the tourist information centre. You'll be able to find maps, coin lockers for your bags/suitcases, and bus timetables up to the viewing caldera. The staff at the info desk will also be able to tell if you can get up to the caldera - it can be out of bounds when it's active, or is ejecting poisonous gas! Giving them a call in the morning to check it's all open (Tel: 34-0751).
Nishi and Higashi caldera viewing platforms
There are 2 places to view the Caldera: The classic Aso viewing platform of Aso Nishi and the slightly more unusual (but I think better) viewing platform of Aso Higashi. There are a few differences here to bear in mind.
- The nishi platform is often very busy, especially on holidays
- The nishi platform is the only one where you can see into the caldera
- The nishi platform has buses that leave JR Aso every 90 minutes ¥470 per person
- The higashi platform has no buses, and is ¥800-1000 by taxi to the bottom of a cablecar that takes you up
- The higashi platform is newer, very quiet (deserted?), and very peaceful
- The higashi platform gives you a better overall view of the caldera (though you can't see inside)
- You can see the nishi platform from the higashi platform
- There are some great walks around the caldera from Higashi to Nishi giving you the best of both worlds.
I personally really recommend you do the Higashi platform -> Nishi platform walk if the weather is good. You'll need some sturdy shoes, but it's an unforgettable experience. The bus from the Nishi platform will also deposit you handily right back at the station.
Beppu or Kurokawa?
While I like Beppu, and you can get there by hopping on a train at Aso, it has become very commercialised. Have you considered going to Kurokawa Onsen? It's one of my most favourite onsen in Japan and is an unspoilt beauty of a location, with wonderful outside open baths, and a very laid back woodland feel. Best of all, there is a bus to get there that leaves from JR Aso direct to Kurokawa Onsen, making it a cinch to access after your day at the volcano. There are only 5 a day, so make sure you check with the Tourist Information desk when you arrive for the bus timetable. The bus is not a JR bus and will cost around ¥900. Also, hotels there can be booked up, so it's best to book your stay in advance. There's a pretty good list to give you some inspiration here