acrivate card in tokyo

Your post is a little confusing - so on the 26th you plan to go to HIroshima (and presumable Miyajima) from Kyoto? My advice if you want to see both Hiroshima City and Miyajima in one day is start EARLY. You'll be able to cover some of the main sights in HIroshima (Peace Park/Museum, Shukkeien Garden, Hiroshima Castle) but seeing Miyajima will take about ¾ of a day.
Spend 1½ days in Kyoto, and spend ½ a day in Nara. Since you are already seeing Tokyo, you can largely skip Osaka during the day. After the temples of Kyoto close down around 5PM, 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 beautiful.

Your next question is when to activate the pass - but are you leaving Japan from Tokyo or Osaka? If you are in Japan for less than 7 days, you should activate it at Narita Airport and take the Narita Express into Tokyo.

Hello Elvira,

Your hotel is very close to Shinjuku station. I would advise activating your JR Pass there :)

You can also do so at the Airport but there's generally a much longer queue.

Hope this helps,
Daniel

...but if you activate it at the airport you can take the Narita Express to Shinjuku, and don't have to pay an additional ¥3,000 yen.