The trains from Narita Airport to Asakusa station runs normally on January first?
Trains in Japan run on New Year's Eve, and January 1st. For most of the day, they will operate with a weekend/holiday schedule; however, some lines expand their service to cover late-night and early morning "hatsumode" visits. I doubt that the airport line sees any increase in service.
If you search hard enough, you may come across some information about Keisei's expanded service at New Years between Narita and Asakusa, as both of those areas are extremely popular for hatsumode. Warning: This is not for Narita Airport, but the not-so-nearby Narita Station. This press release (in Japanese) was made last year on December 4, so it is reasonable to expect that you can get similar information around the same time this year. They may include the airport in their expanded schedule, but I doubt it.