After a suspected poisoning of Alkibiades, you should go to the Sanctuary of Olympia to investigate what has happened at the banquet. Go to the building in the south west of the sanctuary and inspect any items that have search icons associated with them. Once you have discovered the name of the merchant who delivered the poison, pay him a visit. You will find him in a room south east of the banquet area. When you arrive, you will be ambushed by one cultist guard. Take the battle outside of the room to avoid killing Priam and focus on single attacks so that you do not harm any civilians. When the guard is dead, speak with Priam. You can choose to spare him or kill him, but whatever your decision be sure to extract the information you need before you act on it. Priam will inform you about where you can find an antidote to the poison so that you might save Alkibiades; Fort Koroibos, located north east of the room where you were ambushed.
When you arrive at Fort Koroibos, use Ikaros to identify the location of the antidote, and any surrounding guards that you will need to avoid. You can then meditate until nightfall to allow easier infiltration of the fort. The antidote can be found in a tower at the centre of the fort. The easiest way to it is by using the foliage which begins from the south western wall to remain hidden from the view of guards. Crouch through the grass until you reach the tower, and then enter from the gap in the wall to the north east. You will find the antidote in a treasure chest within the tower. Once you have it, leave the same way you entered and return to Alkibiades. When you administer the antidote, the quest will complete.