Athens to Patras
Best would be to take the coach/bus. If I were you I would pick the morning 8:30 Express bus according to the official Patras coach station website. I have no personal experience, but various sources say it takes ~ 3 hours (maybe a bit less).
Patras to Araxos airport
The reason you cannot find any information, is because it doesn't exist :) The aforementioned official website does not list timetables for local routes. But they do serve the airport. Whenever there is a flight, they offer a bus service to and from the airport. The only unofficial website I could find mentioning is here. It takes ~ 1 hour and costs 4 euros per person, per leg (2012 price, might have gone up). Source for this information was a travel forum (in greek).
Hope that helps!