National Museum of Asian Pacific American History and Culture Act This bill establishes a National Museum of Asian Pacific American History and Culture within the Smithsonian Institution to provide for the collection, study, and establishment of programs related to Asian Pacific American life, art, history, and culture. The bill also establishes a National Museum of Asian Pacific American History and Culture Council to: (1) make recommendations to the Board of Regents of the Smithsonian Institution concerning construction of the Museum, and (2) advise and assist the Board on all matters related to the Museum's administration and preservation. The Institute of Museum and Library Services shall establish: (1) a grant program for improving operations, care of collections, and development of professional management at Asian Pacific American museums; (2) a grant program that provides internship and fellowship opportunities at such museums; and (3) a scholarship program for assisting individuals who are pursuing careers or carrying out studies in the arts, humanities, and sciences in the study of Asian Pacific American life, art, history, and culture. The Board shall designate a site for the Museum within 12 months of this bill's enactment.