From the Airport, take S-Bahn S1 towards Wedel, and get off at Hamburg Central Station (Hauptbahnhof). Change across the platform to take S-Bahn S21 or S31, taking you to Dammtor (1 stop) or Sternschanze (2 stops).
From Dammtor station, it is just a 15 minutes’ walk to the Central Entrance of the Hamburg Fair site.
From Sternschanze station, it is just a 10 minutes’ walk to the Central Entrance.