Flood Protection Public Safety Act of 2013 - Authorizes the Secretary of the Army, acting through the Chief of Engineers, to implement any flood risk management project for which the Secretary has transmitted to Congress, before the date of enactment of this Act, a letter that includes: (1) a description of the project as recommended in a report by the Chief of Engineers that concludes that the project is technically sound, environmentally acceptable, and economically justified; and (2) notification that the Office of Management and Budget (OMB) has concluded that the report's recommendations are consistent with the President's policy and programs.