There is JAPANiCAN, which is an advertisement on this screen to the right. It's probably the better value in terms of speed. Just click on the big red "Search" button, and when the JAPANiCAN link comes up select "Tickets and Transport" tab.
There is also the Purrato Kodama discount ticket, which includes a drink:
http://www.jrtours.co.jp/kodama/en/?scid=jtt_kepEng001A unreserved ticket for Tokyo to Kyoto is about 13,000 yen each way. The discount tickets are about 11,000 each way. It does save you money. The better transport in Kyoto is usually the bus system, so the JR Pass just sits there for several days. Nara and Southern Kyoto is the exception. The Japan rail pass is 29,110 yen, so it is more than two tickets, and discounted tickets save you even more.
Most people can usually visit Kyoto in four days, so the question is what you plan to do the other three days? As far as the JR Pass goes, it is more expensive, but say, you decided to see the three best views in Japan on the three extra days.
http://www.japan-guide.com/e/e2250.htmlEach of these can be a day trip from either Tokyo or Kyoto. Matsushima bay from Tokyo is about a 2 and a half hour trip each way, but doable. Amanohashidate and Miyajima are both about two hours away from Kyoto. Each of these trips is about the same cost as the Tokyo to Kyoto trip, so adding this means that you are going to rack up over 80,000 yen in travel costs. The JR pass now works.
Tourism-wise, there is not much to see and do before 10 AM, and since the trains start running about 6 AM, this works. Many tourist attractions close around 4-5 PM, and you can commute at this time back.
You might just want to stay local, and say, go to Universal Studios in Osaka, which can also be done with the JR Pass, but this costs 800 yen each way. The Nara trip is 710 yen each way, so if you are doing only small side trips, then get the discounted tickets.
You will have to plan your trip a bit, to figure out what is best for you.