If a Japanese woman marries and American man does she have automatic dual citizenship?

The answer is no. She would have to get a spouse visa to come to the US and the husband should start the Immigration process for her through sponsorship.
If you remain married:

• After two years of living in the US, she will be eligible to get a green card.
  
• After living in the US for 3 years since getting the green card, she can apply for US citizenship if she meets the residency, language abilities, and criminal background requirements.