Don't waste your time on the shrine, but DO COME to Haneda, Tokyo's best international airport. If you have a layover, there are convenient hotels such as the Toyoko Inn nearby, and if you want to just find a place to sleep in the airport, then that is ok too (rare these days). If you are visiting Japan, then you can shave up to an hour commute to Tokyo with Haneda's easy access and much closer proximity to the city.
This former domestic-only airport is well-equipped to handle international traffic with the new international terminal and since it is still ramping up flights, the lines for customs and immigration are much shorter than Narita.
TIP: When searching for flights to the Tokyo area, type TYO instead of NRT or HND for the airport and the search will find you the best option from both airports. And if a Haneda flight comes up, don't hesitate. It is well equipped to handle you and is more convenient than Narita.
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