score:6
They made a press release in 2010 about this service. The advertised domain for this service, specially targeted at backpackers, does not exist anymore, so I would assume this service is discontinued.
If you are willing to pay a bit more (¥2,700), you can use the Airport limousine bus to bring you from Narita to Asakusa.
Upvote:1
According to http://www.rome2rio.com/s/Tokyo-Narita/Asakusa … there is a train for about $14 US, and a bus for about $25 US.
Upvote:3
The Keisei Skyliner also offers two more options which are cheaper than the Airport Limousine Bus.
They offer from Narita Terminal 1/2 a train to Asakusa for ¥1,280 (about $12.5) on the Narita Sky Access Line and ¥1,090 (about $10) on the Keisei Main Line.
Here's a full PDF map for the same.