Apollo Lunar Landing Legacy Act - Establishes the Apollo Lunar Landing Sites National Historical Park on the Moon as a unit of the National Park System.

Requires the Secretary of the Interior and the Administrator of the National Aeronautics and Space Administration (NASA) to complete jointly a general management plan for the Park.

Authorizes the Secretary to enter into one or more agreements with the head of a federal agency to provide public access to, and management, interpretation, and historic preservation of, historically significant Apollo lunar landing site resources under that agency's jurisdiction or control.

Directs the Secretary and the Administrator to enter into an agreement providing for the primary management of the Park by the Administrator.

Requires the Administrator's responsibilities to include: (1) ensuring proper monitoring of the Apollo lunar landing sites; (2) managing access to the sites, including through coordination with other spacefaring nations and entities; and (3) in conjunction with the Director of the Smithsonian Institution, ensuring an accurate cataloging of items in the Park.

Instructs the Secretary to submit the Apollo 11 lunar landing site to the United Nations Educational, Scientific, and Cultural Organization (UNESCO) for designation as a World Heritage Site.