You are not including your full itinerary, so it's impossible for me to say if the rail pass will pay off for you - are you getting a 7 or 14 day? Did you run the numbers for regular fare through a site like www.hyperdia.com to see if the rail pass is worth it? Are you returning to Fukuoka for your homeward flight?
You can certainly see Beppu for a day. The hot springs are the most famous attraction, but there is also the Hells Tour and nearby monkey park/aquarium which are quite good as well.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-hOggZamc2lUNTr46ApKX...Yufuin is more hot springs. Either more relaxing or more dull - depends on the person. There is some quaint shopping and hiking there though. You might consider going to Kurokawa instead if you are going for the hot springs.
Your next stop is somewhat vague. There are many mountains among the Kuju Mountain range - where specifically are you going? And if you want to get to Takachiho after that, the most logical way would be by Kyusanko bus down to the Aso area and then catch another bus. This info might help you:
http://www.japan-guide.com/e/e4775.htmlBut be aware that you'd also need to get down to Takamori Stn to change buses - another headache.
Going to Kuju as a day hike from Fukuoka would be rather tiring - close to 3 hours total by train and then bus. Under your plans, renting a car actually makes a lot more sense if you can drive.
Getting from Takachiho to Kagoshima is not much fun either. The fastest way would be taking a bus to Nobeoka and then a train the rest of the way - best time is about 3 hrs 15 min.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-irtXXB8IJMdkB9mF0qjn...With a good connection, you can get from Kagoshima to Nagasaki in about 3 hrs or a bit more. Take the bullet train to Shin-Tosu and change trains for Nagasaki. It really is the most interesting city on the island. Be sure to eat the chanpon and saraudon while there.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-gbw2yXNLK92siHyNqSNx...For Gunkanjima though sometimes the sea is rough and the boat trips there are cancelled for the day. So I suggest you try it on your first day, and if you can't then you still have the next day to try again.
For your last day, you should be able to cover a number of good places in a full day. The best though is the Nanzoin Temple about 20 minutes by train from Hakata Stn. Just magnificent.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-j4RLa6iC7ayvBAbOn8-0...For other places, you might look into getting a subway day pass or a Fukuoka Tourist City Pass
http://yokanavi.com/access_tourist_eg
