JR Narita Express [NfEX] trains do not run to Ueno Station.
Keisei Skyliner trains and some other Keisei trains run to Keisei Ueno Station.
Keisei Ueno Station and subway Ueno Station (on Tokyo Metro Ginza Line) are located apart from each other; one underground passage connects the two stations, which is fully equipped with elevators.
what is the easiest and cheapest way to go to Asakusa Kaminarimon subway station? Generally the cheapest way of transportation is not the easiest way.
Which terminal of Narita Airport will you arrive at?
What place near subway Asakusa Station will you head first to?
(There is no station with "Kaminarimon" in the name.)
How many travelers will you be?
One option is to go first to Ginza by the Access Narita bus (1000 yen per adult), walk a little then take a subway train on Tokyo Metro Ginza Line from Ginza (G-09) to (G-19) Asakusa.
Actually there are two subway Asakusa Stations.
: Asakusa Station (G-19) on Tokyo Metro Ginza Line.
: Asakusa Station (A-18) on Toei Subway Asakusa Line.
- Map (in Japanese) of subway Asakusa Stations
http://www.tokyometro.jp/station/yardmap_img/_station_%E6%B5%85%E8%8D%...In Tokyo Metro Asakusa Station (G-19),
which is the east terminus of Ginza Line,
the exit which is closest to Kaminarimon is Exit 1.
When you get off a train which arrives at Track 2 (#),
you can pass through the nearby gate, walk on the passage and go up by elevator to this Exit 1.
#: On the body of Ginza Line train cars, you see signs indicating the track number of Asakusa Station.
In Toei Subway Asakusa Station (A-18),
the exit which is closest to Kaminarimon is Exit A4.
However, as you see in the station map,
there is a chain of staircases between the Toei Subway concourse and this Exit A4.
When you avoid staircases and choose another exit, you may hope to take a taxi.
- Tokyo Subway Route Map
http://www.tokyometro.jp/en/subwaymap/pdf/rosen_en_1702.pdf: Also some non-subway sections appear on this route map.