Technically speaking it falls under the "Long-term Residence" status or if they are the child of a Japanese national "Spouse of Child of a Japanese National" status. Officially speaking there is no "Nikkei Visa".

"Long-term Residence" is not to be confused with Permanent Residency which is another status. Those with a "Long-term residence" status is usually granted for three years, and can be renewed.

A research of the older MOFA website which was much better then the new one they have(the old one is cached via wayback machine) shows the following requirements for "Long-term Resident":

1. A copy of the family registration, the marriage certificate, the birth certificate and other documents certifying the status of the foreign national concerned.
2. Documents certifying that the person concerned can defray all the expenses incurred during the stay in Japan. In cases where the expenses incurred during the stay in Japan is defrayed by another person, documents certifying the income of the person who intends to defray all the expenses.
3. A letter of guarantee by the person living in Japan.

The requirements for "Spouse or Child of a Japanese National":
In cases where the person concerned is a child or an adopted child of Japanese national, the following materials are required.

1. A copy of the family registration of the Japanese national concerned, the birth certificate of the foreign national concerned, and other documents certifying the personal relationship between the parent and the child.
2. Documents certifying the profession and the income of the foreign national concerned or his or her father or mother.
3. A letter of guarantee by the Japanese national concerned or another person living in Japan.

Though from my understanding it's quite hard to get the status (long term resident based on Japanese ancestry). Though any official inquires should of course go through the Japanese Embassy/Consulate you plan on working with.