The museum is a few minutes from Ōmiya Station, in Saitama Prefecture.
This is easily reachable from Tōkyō by using the Keihin-Tōhoku, Saikyō, Shōnan-Shinjuku, Takasaki-Utsunomiya lines.
For those coming from the Tōhoku region by Shinkansen, Ōmiya is on the way to Tōkyō.
Once in Ōmiya, take the Ina line (also known as New Shuttle) and get off at the first stop, which is Tetsudō Hakubutsukan Station.
You can access the museum by using your IC card (like Suica, Icoca, Pasmo etc) for 1000 yen.
Have a look at the official website: http://www.railway-museum.jp/en/