The prime reason for the difficulty in getting seats is the general lack of service between the USA and Athens. Only two direct daily flights---DL/JFK and AA/PHL.....plus a second DL trip from JFK. That's it.....despite the fact that the number of Americans visiting Greece has increased. So mostly dependent on the main European hubs.

Go to the Parthenon museum. It had been re-done a couple of years back. You will get a better appreciation and understanding of the Acropolis.

Lykavitos is a spot that is often overlooked by US tourists. This is the "other" main hill in Athens. At the top, you get the best view of the city.
Also, there is this unique kind of train that goes inside the hill, to get to the top.

The metro system works really well, in getting you from the city to the airport. However, if you are in the first group of flights in the morning out of ATH, that can pose a problem. If I recall correctly, the trains don't start up until 530am....and the airport express bus service is limited at that time, too. So plan accordingly.