Monuments Protection Act This bill directs the Departments of the Interior and Agriculture (USDA) to enter into an agreement with a state, the District of Columbia, a federally recognized Indian tribe, a U.S. territory or possession, or a political subdivision that offers to assume activities to resume or continue operations and public access to any covered unit when those activities would not otherwise be carried out: (1) because there is a lapse in available federal funds as a result of a failure to enact appropriations, or (2) because of a failure of the federal government to make sufficient funds available to operate such unit. The bill defines a "covered unit" to mean: public lands; units of the National Park System, the National Wildlife Refuge System, and the National Forest System; federally owned open air monuments and memorials; and federally owned roads and off ramps on all such lands. The bill requires Interior and USDA to reimburse an entity for such activities once funds are made available.