One simple route is
Narita Airport Terminal 1 / Terminal 2
| {Narita SKY ACCESS}
(Takasago
: If your first train is NOT bound for Ueno, change trains here at the same platform.)
| {Keisei Main Line: train bound for Ueno: about 10 to 20 minutes}
Nippori -transfer gate- Nippori of JR-East: Track 4
| {Joban Line: any train departing at Track 4: about 5 minutes}
Minami-senju.
It takes about 80 minutes. One-way fares are 1350 yen in total and covered by Suica or PASMO.
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http://www.japan-guide.com/e/e2359_002.htmlPlease note that (Keisei) Narita Station and Narita Station (of JR-East) are outside the airport.
How is Minami-senju convenient for getting around Tokyo during the day, and especially during the evening/night? That may depend on places to visit.
Minami-senju Stations are on JR Joban Line, Tokyo Metro subway Hibiya Line and non-JR non-subway Tsukuba Express Line.
On each of the three lines, trains are available to Minami-senju until around the midnight.
All of northbound trains departing from Minami-senju cross over Sumida River and make a stop at next Kita-senju Stations, a big terminal.
From Minami-senju you can get to Asakusa via Kits-senju or via Ueno. Also Tsukuba Express Line has Asakusa Station, which is located apart from other Asakusa Stations.
You can get via Nippori Station to stations on Yamanote Line such as Ueno, Tokyo, Shinagawa, Shibuya, Shinjuku and Ikebukuro.
By subway Hibiya Line you can get to Ginza Station, Kamiyacho Station (near Tokyo Tower), Roppongi Station, etc.
Fares on Tsukuba Express Line are relatively high, but you can change lines in Shin-okachimachi Station to Toei subway Oedo Line and get to Tochomae Station (near Tokyo Metropolitan Government Buildings).