How early do we have to arrive at the airport?

Since this is a domestic flight you don't have to get to the airport extra early, however you still need to get there to check in and go through security. I'd said be at the airport no later 3:30 a.m.

The cheapest way to get to the airport from the center of Athens is by the Metro. Unfortunately they don't start until 5:30 a.m. so that's out. The next best, but the most expensive is taxi. The earlier in the morning the more expensive. I would think around 40-50 Euros. I don't know what the bus schedule is out of Athens and not sure what time they start running. Hopefully there is someone who does, but it appears a taxi would be the most convenient but most expensive. But if you are splitting the cost then it won't be that bad.

The airport bus #X95 to/from Syntagma Square runs 24 hours/day; see www.oasa.gr. Athens airport is relatively small and easy to navigate, but the boarding procedure may take longer, such as a shuttle bus out to board the plane.

The Metro is NOT the cheapest way to get to the airport. The X95 Airport Express Bus is only €5 and leaves Syntagma Square 24/7/365 every 20-30 minutes. That early in the morning it will take abut 45 minutes to get to the airport, which is about the same time it takes the Metro to cover the same distance. Buy your tickets from the driver of the bus. Have exact change. Validate the tickets inside the bus in the ticket machine. If your flight leaves Athens at 5am that means you need to catch a bus at Syntagma Square no later than 3am. If you're flying on Aegean Airlines or Olympic Air you can print out your boarding passes online 48 hours in advance of your scheduled departure, so when you get to the airport all you have to do is drop off any bags that need checking and enter the security area to find your departure gate. There's a special drop-off area for this.