There's shower and dayroom rental in NRT. 30 min of shower for 1,000 JPY and 1 hour of day room for 1,500 JPY (They have raised the price

). This is probably the cheapest option.
http://www.narita-airport.jp/en/guid...svc_37.html#t2
As booking class E is not available on your flight. You will have to book a Y normal fare which probably costs almost the same as the cheapest C ticket.