Really there is no answer for this. Just before your long distance travel starts is the obvious choice, but you need to give more info on how much time you are planning to spend in each place.
Well, there are 2 ways you can do this. You can either get a 7 day pass or a 14 day pass. Both would pay off for your plans, but you would get greater value doing it as a 7 day pass. To do that, you'd need to do Hakone as a side trip from Tokyo. You can get a Hakone Free Pass - a JR Pass can't help you see Hakone anyways. http://www.japan-guide.com/e/e2358_008.html http://www.japan-guide.com/e/e5210.html
Then you use the 7 day pass to go from Tokyo to Takayama, Kyoto and Hiroshima, then back to Tokyo within 7 days.
For Kyoto, you should not miss Nara either - it is worth at least half a day, and for Hiroshima, don't miss Miyajima. You can also use the pass on the Hiroshima City Loop Bus to get around, as well as the JR ferry to Miyajima. Since you are already seeing Tokyo, you can largely skip Osaka during the day. After the temples of Kyoto close down around 5 PM, you can zip over to Osaka for the evening. At night the city comes alive and has some great places to see, such as Dotonbori, plus the night views from the Umeda Sky Bldg and Abeno Harukas Bldg are wonderful. On another evening, you could also go and see Kobe.
Comments
Tokyo - 5 days
Hakone - 2 days
Takayama - 2 days
Kyoto - 3 days
Hiroshima - 2 days
Tokyo - 2 days
To do that, you'd need to do Hakone as a side trip from Tokyo. You can get a Hakone Free Pass - a JR Pass can't help you see Hakone anyways.
http://www.japan-guide.com/e/e2358_008.html
http://www.japan-guide.com/e/e5210.html
Then you use the 7 day pass to go from Tokyo to Takayama, Kyoto and Hiroshima, then back to Tokyo within 7 days.
For Kyoto, you should not miss Nara either - it is worth at least half a day, and for Hiroshima, don't miss Miyajima. You can also use the pass on the Hiroshima City Loop Bus to get around, as well as the JR ferry to Miyajima.
Since you are already seeing Tokyo, you can largely skip Osaka during the day. After the temples of Kyoto close down around 5 PM, you can zip over to Osaka for the evening. At night the city comes alive and has some great places to see, such as Dotonbori, plus the night views from the Umeda Sky Bldg and Abeno Harukas Bldg are wonderful.
On another evening, you could also go and see Kobe.
You can see previews of some of the best places to see on:
https://www.youtube.com/user/thejapanfaq/videos
we already booked the hotels so we can't change it
is it possible to activate the JR pass from Takayama station?
No, but you could from Nagoya. The complete list is here:
http://www.japanrailpass.net/en/exchange.html