When I flew to Egypt I used a miles ticket for U.S. to Athens and then bought a ticket from Athens to Cairo direct on Aegean airlines. It was around $200 - $300 dollars but worked out well for me. I also was interested in seeing Athens as well. I'd recommend a one to two day stopover to see the sights there. The city doesn't have as much to see as London or Rome so I think you can see the highlights in 2 days.