There are post offices all over Athens; ask your hotel how to get to the closest one. The city's two central post offices are in Syntagma Square and off Omonia Square; both are infamous for long lines and slow service. Avoid all post offices during the first week of the month, when Greeks line up to pay their utility bills. Post offices are open weekdays 8-2; some also open Saturday morning. If you want to mail a letter, you can do it from the yellow mailboxes outside post offices; there are separate boxes for international and domestic mail.
Omonia Square. Syntagma Square.