If you activate your JR Pass to use from the 18th, it would expire at midnight at the end of the 24th. A day starts and stops at midnight, so your first "day" is just 12 hours, from what you write. As for getting back to Narita airport, the cheapest is by bus:
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 wonderful. Instead, you might go see Himeji Castle, which is Japan's finest and newly renovated. It's also unclear why you'd go to see Nagoya - it's the most boring of all the big Japanese cities. Nearby Inuyama Castle is nice, but if you want to see something nice, go farther to visit Takayama and Shirakawago, to see some old style dwellings that are still very much in use.
And since you are spending 4 days in Tokyo, and really 3 is enough, on your last day of your Pass you could take a day trip to see Nikko which is a fabulous place to visit.