7 or 14 JR Pass

My itinerary for my upcoming trip in Japan is as follows:

8th to 12th August - Arrive at Narita Airport, transfer to hotel in Shinjuku. Mostly travel within Tokyo.
12th to 15th August - travel to Mt Fuji, stay at hotel in Funatsu. Hoping to go to Fuji q land, stay in onsen, and half climb on Mt Fuji
15th to 20th August - travel to Osaka, stay at hotel in Chuo Ko. Maybe day trip to Kyoto and Nara. Departure flight from Okami Airport to Narita then back home

Just wondering would it be cheaper if we get just the 7 JR Pass, and activate on 12th August when we leave for Mt Fuji, and then just pay a one way fare from Narita airport to get to our hotel on Shinjuku (i.e., either train or car, airport shuttle). For any other travel that isn't covered by the JR Pass, we'll just use the metro or suica cards?

Have I got everything right? Just trying to be cost effective!