Federal Aviation Administration Fair Labor Management Dispute Resolution Act of 2006 - Amends federal transportation law to revise certain federal aviation administration personnel management system requirements to prohibit the Administrator of the Federal Aviation Administration (FAA) from implementing any proposed change to the FAA personnel management system in cases where the services of the Federal Mediation and Conciliation Service do not lead to an agreement between the Administrator and FAA employees unless a bill is enacted into law by Congress that authorizes the change during the 60-day period beginning from the time Congress received the proposed change. (Current law does not require congressional action.) Requires the Administrator and FAA employee bargaining representatives to submit the proposed change to binding arbitration if Congress does not enact a bill into law within the 60-day period.