Length of Journey

Hi,
It depends on how you travel. As for paying regular fare on the fastest bullet train (Nozomi), it takes about 150-155 minutes at a cost of ¥13,620 (unreserved seat). Taking a different type of bullet train is typically about 30 minutes longer, at the same fare.

You can see regular fares and schedules on Hyperdia.

If you're only going one way, a rail pass probably won't help you unless you travel way farther than Osaka. Still, some regional passes might help. See here.
You could spend a lot of time in Tokyo and still not see everything.
In Tokyo a few of the best places to see are the Tsukiji Fish Market, plus the site that every last tourist goes to see, the Sensoji Temple. If you are in Tokyo on a Sunday afternoon, be sure to go visit Harajuku to see the youth with their wild fashions. Meiji Shrine and Yoyogi Park are next to it and well worth seeing also. And be sure not to miss the Shibuya Hachiko intersection especially on a weekend when it looks like half the city is crossing the street all at once.
In Kyoto you could spend weeks there and not see everything. The three best sights though are the Kinkakuji Temple, Kiyomizudera Temple, and Fushimi Inari Shrine. Fushimi is the most time consuming; you could spend a couple of hours to over half a day if you want to romp over all the trails. And the Otagi Nenbutsuji is one of the most humorous and whimsical places you could visit in Japan - well worth visiting, even for kids.
Nara is often neglected by some tourists, which is a shame. Missing the Todaiji Great Buddha in Nara is like going to India and skipping the Taj Mahal. Nara Park has a lot of great places, such as Kasuga Shrine, Kofukuji Temple, and feeding the many deer in the area.
A few other great places missed out even by those who go to Nara though are the Isuien and Yoshikien Gardens. They are gorgeous and not crowded at all.
In addition, there are other numerous great places you can visit in the area. Okayama is not far and it is famous for its castle, plus Korakuen, one of Japan's Top 3, and also the Handayama Garden. If you'd like to get off the beaten trail for a while, the half day hike up to Konpirasan in Kotohira is stunning. Going to Takamatsu is also nice - it has one of Japan's finest gardens also, called Ritsurin.

Hiroshima also has a number of good places. Miyajima is the most famous, and aside from Hiroshima itself, Onomichi with its temple tour and Kosanji Temple is spectacular. Just beyond Miyajima also lies Iwakuni with its historical bridge and castle on top of the mountain. Taikodani is also a beautiful shrine in Tsuwano, Shimane. Hagi Castle was actually delightful, and the preserved samurai district and Mori related temples a rarity. One of the very best is the Akiyoshi Limestone Cave, a 300 million year old wonder with huge caverns and underground river running through it.

If you can, try to find a little time to see the Japanese themselves. Go visit a Japanese supermarket and see what people typically eat. Look through a park and watch the Japanese kids play. Learn about the history of some of the places you visit before your departure - it will give your trip so much more meaning.

Good luck.