Artificial Reef Promotion Act of 2013 - Amends the National Fishing Enhancement Act of 1984 to require the Secretary of the Army, in issuing a permit for artificial reefs, to ensure that a state assuming liability for future damages has established an artificial reef maintenance fund. Requires the Director of the Bureau of Safety and Environmental Enforcement and the Director of the Bureau of Ocean Energy Management to promulgate regulations that expedite the review of a final application submitted to dispose of or remove an offshore oil and gas platform in the Gulf of Mexico for use as an artificial reef by requiring that a decision be made within 150 days after the submission of such application. Requires the Commanding General of the Corps of Engineers to promulgate regulations that expedite the review of a final application by the Secretary by requiring a decision to be given within 120 days after the submission of such application. Directs the Commanding General to designate no fewer than 20 artificial reef planning areas. Specifies location and depth requirements for such artificial reefs, including the number of areas that should be located outside the seaward boundary of each of the Gulf states. Revises siting compliance requirements imposed upon permittees. Prohibits regulations from requiring an artificial reef planning area to be filled to capacity with offshore oil and gas platforms in the Gulf of Mexico before another artificial reef planning area is established. Requires the Regional Supervisor to give preference to a final application submitted to dispose of or remove a platform for use as an artificial reef.