The closest hotel to the airport is the Sofitel. It's a 5-minute walk from the Main Hall. But it's very expensive. The Attica Avenue Holiday Inn is about 20 minutes away and they have a shuttle bus. I don't know if it operates so early in the morning, you'll have to contact them and ask.
The St. Thomas B&B and Peri's Hotels are not far away either.
With these two you'd have to enlist the assistance of the owners to make sure a taxi was there to take you to the airport around 3 AM. If you stay in Central Athens you can get a room near Syntagma Square and take the Airport Express Bus #95 from alongside Syntagma. It leaves every half-hour 24/7/365. Check the Pan, Arethusa, Amazon and Metropolis Hotels.
The airport is one of the least sleep-friendly I've seen but there's an upstairs area just outside the chapel that's quiet. Taxis cost double between midnight and 6 AM.