The Japan Rail Pass allows travel on Japan Rail lines. Inside the city (I assume you mean Tokyo,) the Pass allows travel on all JR lines within the city. That includes the Yamanote Loop Line around the city and several other lines in and around the city area. It does NOT cover the vast subway system or the various private rail lines.

If you are using a Japan Rail Pass for transportation to and from Narita Airport, the best option is the Narita Express Airport train. From Narita to Tokyo central station, the trip is closer to an hour or a bit more, depending on time of day.

Although the JR Rail Pass does work on some of the trains in the city, your best option for getting around Tokyo is probably the Suica pre-paid travel pass. It is useful on the subways and rail lines around the city and can also be used to make purchases in many stores and businesses. The Japan Rail Pass will get you to most places in the city. The Suica Pass will get you everywhere in the city.

If you plan to be in Japan less than 7 days in total, a JR Rail Pass could be used for both the Narita Express into and out of the city on your arrival and departure days. If you only use the Pass for those two trips and a round trip to Sendai, you don't quite get your money's worth. The 7-day Rail Pass costs 29,100 JPY. The 4 trips listed would cost you approximately 27,000 JPY if purchased separately. If you add another day trip or two, say to Nikko or the mountains, the Pass would pay for itself. If you don't do any other rail travel except local trips within Tokyo, it MIGHT just break even since you could just walk on the local lines by showing your Rail Pass.

Have you looked into spending those few days outside of Tokyo in Kyoto, Nara or Osaka instead of Sendai? The site-seeing and historical areas in the Kyoto-Nara-Osaka area are much superior to what you will find in Sendai. If you can find a reasonable hotel in one of those cities, I highly recommend them over Sendai for a first time visitor.

If you were to use the Rail Pass for both directions on the Narita Express and a round trip to/from Kyoto, the Pass would definitely save you money.