Confusion how JR pass works

Hello,
So my brother and I will be arriving to Japan (Haneda Airport) on September 11th around 1 p.m. We planned to go to Kyoto on the same day we arrive and stay there for a couple days. So my question is how to does the 7 days JR pass work? From my understanding is once you activate your pass on the first day is counted as a whole day so pretty much only 6 days?. So my concerns is will it be enough to cover our trips. Here is our itinerary:

Sep (11th) Day 1: Haneda airport/Kyoto
Sep (12th) Day 2: Kyoto
Sep (13th) Day 3: Kyoto
Sep (14th) Day 4: kyoto
Sep (15th) Day 5: Kyoto/Nara
Sep (16th) Day 6: Osaka
Sep (17th) Day 7: Go back to Tokyo
Sep (17th - 24th) : Tokyo

Will the 7 days pass cover it or is it even worth it?

Thanks!