The Athens suburban railway connects the Athens airport with Corinth, a town at the top of the Peloponnese, across the isthmus from the continental mainland, about 90 minutes from Ermioni/Metoxi. The trip requires a train change: you board at the airport, change at the Ano Losia station, and continue on to Corinth. After arriving in Corinth, you can take a bus to Kranidi or Ermioni or hire a taxi to take you to either of those places or to Metoxi. For information and timetables, visit the Athens Airport Bus website.
The central bus station in Athens runs buses to Kranidi and Ermioni. From either you can take a taxi to Metoxi; from Ermioni you can take a boat to Hydra and will find overnight accommodation if needed. Please note that Metoxi offers no overnight accommodation.
The bus routes and times are listed in English on the KTEL website. This is a fairly long trip and not as convenient as catching the hydrofoil from Pireaus, but it’s a handy option when boats aren’t running due to either weather or strikes. The buses are new and very comfortable and the mountain roads south of Corinth on the Peloponnese are scenic and often breath-taking, if at times a bit harrowing.