I am planning to come to Osaka morning of 11/19 and go straight from the airport to miyajima island. Go back to Osaka on 11/22 afternoon and planning to travel back and forth between Osaka and Kyoto from 11/22 to 11/28. Can you recommend which pass or JR pass would fit my plan.
Please note however that you cannot use the bullet train between Osaka and Kyoto with that pass.
For the rest of your time, the fare is not that much between Kyoto and Osaka, but if you want to save some money, you could use the Hankyu Tourist Pass.
It goes to the Karasuma area of Kyoto, what some call the 'downtown'. If you want to go to Kyoto Stn however, it would be better to use a JR train. There really is no JR regional pass that would pay off for just going between Kyoto & Osaka. If you have the time and the chance, spending at least half a day to see Nara is also very worthwhile - it has some of Japan's top sights.