Hi Omar,
Yes, you can use the Japan Rail Pass on the Narita Express which runs between Narita Airport and Tokyo, and you can also use it on the Sakura or Hikari bullet trains between Nara/Kyoto and Hiroshima.
About whether a rail pass makes sense for your trip, lets take a look at your costs:
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Narita airport terminal 1 > Tokyo: ¥3140, (62mins, 79km)
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Tokyo > Kyoto: ¥13420, (164mins, 513km)
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Kyoto > Nara: ¥690, (45mins, 41km)
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Nara > Hiroshima: ¥10990, (167mins, 393km)
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Hiroshima > Kyoto: ¥10990, (126mins, 380km)
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Kyoto > Tokyo: ¥13420, (161mins, 513km)
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Tokyo > Narita airport terminal 1: ¥3140, (58mins, 79km)
TOTAL: ¥55790 for 7 legs if purchasing JR tickets
Whether a 7 day pass is worth it will depend on if you decide to visit Hiroshima or not. In general a 7 day pass (¥28,300) starts to break even if you do a return trip to Kyoto. In your case however your return to Tokyo would be out of the 7 day period and so a 7 day pass will not be worthwhile for your itinerary in it's shortest state.
However, if you were to do a return trip to Hiroshima, your standard JR ticket costs for the period would be ¥36090 within a 7 day period and you would save ¥7790 with a 7 day pass.
If you do visit Hiroshima, you can save even more with a 14 day pass. Including the return to Tokyo and the travel to and from the Airport into your overall itinerary, you would make overall savings of ¥10,690 overall with a 14 day pass. In addition to these base savings you have the flexibility to change your plans (eg, making a day trip to Nikko from Tokyo), and can make good use of the JR Pass around Tokyo and to a limited extent in Kyoto to make further savings. Here's a set of maps of JR lines around major cities that should help you plan.
Hope this helps!