Does the ship you are sailing on just dock in the port of Pireaus while visiting the mainland of Greece? Or, will the ship also be docking in the port of Itea which is near Delphi? If you are just docking in Athens I would stick to the ships excursion because of the logistics. Key tours or Chat tours might have a six hour trip to Delphi but they may also be the same companies that the ships would use for their excursions. If the ship is docking in Itea you can take a taxi to delphi about a half hour away or you can take the local bus from Itea to Delphi. The ktel buses also operate from athens to delphi but the logistics of getting back to the ship might be an issue prior to the ship sailing on to their next port of call.
Adventure of the Seas (RCCL) 3/10 -Southern Caribbean
Adventure of the Seas (RCCL) 3/07 -Southern Caribbean
Voyager of the Seas (RCCL) 3/03 -Western Caribbean
Monarch of the Seas (RCCL) 3/01 -Southern Caribbean
Song of America (RCCL) 10/90 -Western Caribbean
MTS Jupiter (Epirotiki) 7/84 -Mediterranean
As far as I remember a tour to Delphi out of Athens cannot be done whithin 6 hours. Dnot forget you also need time to visit the place.
As mentioned if offered you should book a shipexcursion in this case, so if there is of delay the ship will wait for you.