No specifications on times of the day, day of the week, and season. Most flights from the mainland US are scheduled to arrive at Narita around 1600 and depart around 1800. With your saying a 12-hour layover, I believe you would land in Narita around 600 from somewhere. After the arrival, you would be ready to visit Tokyo around 700 at Narita. You would have to come back to Narita by 1600.
I could nominate Kokyo (Tokyo Imperial Palace), Gigido (National Diet Building), Tocho (Tokyo Metropolitan Government), Meiji Jingu (Shrine), and Sensoji (Temple).
I created the below itinerary from the end, Narita at 1600. That resulted in the start from 600. So, you might have to adjust somehow.
Narita 600-(Limousine Bus)-715 Tokyo
From Tokyo to Nagatacho, you would see the JR Tokyo station red-brick building, Kokyo (Imperial Palace), Tokyo skylines, and Gigido (National Diet Building). If you come to Tokyo station by Limousine, you need to go to the opposite site of the station. The path is the most north route correctly seen in the Google Maps below.
https://www.google.co.jp/maps/dir/35.6807287,139.7687668/35.6833448,13...It would be a 3.3-km, 40-min walk. You could only walk between Sakuradamon and Nagata-Cho, by using a train route of Tokyo-(Yamanote Line)-Yurakucho-(Yurakucho Line)-Sakuradamon. But, it would be also tough. It is in a rush hour. Anyway, you need to use Nagatacho station. It could be a maze. You would come down from the gate 1, near the Diet.
http://www.tokyometro.jp/station/yardmap_img/_station_%E8%B5%A4%E5%9D%...Nagatacho 820-(Hanzomon Line)-840 Shibuya
If you come out of the gate 6, there should be the famous Shibuya Crossing. You just cross it to take a JR line to Harajuku. So, you might not have to count on half an hour here.
http://www.tokyometro.jp/station/yardmap_img/_station_%E6%B8%8B%E8%B0%...Shibuya 911-(JR Yamanote)-914 Harajuku
It is the Harajuku, but your aim is just Meiji Shrine. Shrines are OK in the early morning. Go through the Torii gates and walk on pebbles. You pass through the shrine to the opposite side to take a bus to Tocho (Tokyo Metropolitan Government). Sangubashieki bus stop is just under the highway structure. Cars run the left-hand side in Japan. You have to wait for the bus stop at the west side here.
http://www.meijijingu.or.jp/img/access/naien.pdfhttp://busmap.info/busstop/78233/Sangubashieki 1015-(To Bus Juku#51)-1020 Junisoikenoue
Get off the bus once it reaches the Shinjuku Chuo Koen (Central Park). The Junisoikenoue bus stop is near the park. You could pass through the park to get to Tocho.
http://busmap.info/busstop/78235/The south building of the Tocho would be better to see Tokyo. Tokyo is more developed in the southern part up to Yokohama. Tochomae station is just connected to Tocho. You transfer to Yurikamome line at Shiodome. The Yurikamome line goes through Odaiba, the site of the Olympic Games Tokyo 2020. You could see Tokyo Tower as well.
https://tokyo2020.jp/en/games/venue/olympic/Tochomae 1128-(Odeo Line)-1148 Shiodome 1156-(Yurikamome)-1226 Toyosu
http://www.yurikamome.tokyo/file/pc/en/pdf/route_guidance/railroad_map...It would be a 7-min walk from Yurikamome Toyosu to Toyosu pier. If you want to skip Odaiba, you could get on HOTARUNA at Hinode Pier.
Hinode 1225-(Tokyo Cruise HOTARUNA)-1300 Asakusa
Toyosu 1240-(Tokyo Cruise HIMIKO)-1315 Asakusa
http://www.suijobus.co.jp/wp/wp-content/themes/tokyocruise/pdf/timetab...http://www.suijobus.co.jp/en/price/You could enjoy a view of Tokyo Skytree from the Tokyo Cruise Asakusa Pier. Then, I suggest a route of Kaminarimon - Nakamise-Dori - Sensoji - Hana-Yashiki. Hana-Yashiki is the oldest theme park in Japan. You could take a Skytree Shuttle or a To-Bus to catch a Skyliner train that runs on the abandoned Narita Shinkansen route.
http://www.keisei.co.jp/keisei/tetudou/skyliner/us/ae_outline/index.ph...Asakusa-3-Chome 1445-(To-Bus To#08)-1459 Nippori 1505-(Keisei Skyliner #39)-1543 NRT
http://busmap.info/busstop/79250/http://busmap.info/route/21654/Asakusa-3-Chome 1435-(Skytree Shuttle)-1452 Tokyo-Skytree-Town/Oshiage 1501-(Keisei Sky Access)-1557 NRT
http://www.tobu-bus.com/en/skytree/ueno_asakusa.htmlhttp://www.tobu-bus.com/en/skytree/pdf/timetable_uenoasakusa.pdf?20161...Asakusa 1458-(Keisei Sky Access)-1557 NRT
For an early morning, you could start from Tsukiji Fish Market. For your boys, Ghibli could be a good wrap-up site. In between, I set Edo-Tokyo Museum in below. You might see Shinjuku Skyscrapers from the Sobu line and the Shibuya Crossing just before NEX. Don't miss a small Benten Shrine in Inokashira Park.
NRT 710-(Keisei Sky Access)-816 HigashiGinza
Tsukiji Market
Tsukijishijo 928-(Metro Oedo)-944 Ryogoku
Edo-Tokyo Museum
Ryogoku 1059-(JR Sobu)-1141 Mitaka 1150-(Ghibli Bus)-1155
Ghibli Museum
Inokashira Park
Kichijoji 1424-(Keio)-1440 Shibuya 1446-(NEX #35)-1557 NRT
