The discussion of whether or not Hunting is then the oldest sport we do not want to start here. Hunting was first a necessity of life then became an occupation for the nobles and is now a sport and passion for many.
What is the oldest method of hunting is well known. Not Falconry or Falconry, but Coursing, hunting with dogs, deserves this honor. However, on the European Continent, hunting with dogs on live animals is prohibited.