From Athens International Airport to Patras you can travel in two ways:
1. By Bus.
If you choose to travel by bus, you have to do the following:
First, you should catch the bus (line X93) from the airport to «Kifissos» bus station. You can buy tickets in the bus. The ticket costs approximately 5 Euros. Timetables can be downloaded from http://www.oasa.gr/xmap.php?id=px93. From Kifissos Bus Station, you get the bus to Patras. The ticket costs approximately 20 Euros and can be bought on site. There are buses to Patras every half an hour from 05:30 to 22:30. It is a 3 hour road, unless you catch an Express bus, which by and large takes 2 and a half hours. The timetable for the Express bus (Athens to Patras) is: 09.30, 11.30, 12.30, 13.30, 14.30, 15.30, 16.30, 17.30. More detailed timetable: http://www.ktelachaias.gr/?module=default&pages_id=23
2. By suburban railway train –Bus.
If you wish to travel by bus to Patras from Athens International Airport, you can take the bus E93, which  leaves  from  the  main  building  of  the  airport  (ports  4-5)  and  get  off  Kifisos  bus station from where you change bus heading to Patras. From Athens International Airport "Eleftherios Venizelos" you can reach Patras by buses and trains of OSE (public Greek railways) as well. Take the Athens Suburban Railway to Kiato and then take the bus that leaves every 1hour from Kiato heading to Patras. If you choose this option, you have to catch the suburban railway train (PROASTIAKOS) to Kiato Station and then there is a connection with a bus (OSE) that will take you in Patras. The ticket, which covers the whole trip to Patras, should cost approximately 17 Euros and can be bought at the airport. Keep in mind that this is not a direct line and you will have to change at Kiato Station.
Timetables from/ to airport to/from Kiato can be downloaded from http://www.trainose.gr/wp-content/uploads/2014/03/%CE%94%CF%81%CE%BF%CE%BC%CE%BF%CE%BB%CF%8C%CE%B3%CE%B9%CE%B1-%CE%91%CE%B5%CF%81%CE%BF%CE%B4%CF%81%CF%8C%CE%BC%CE%B9%CE%BF-%CE%9A%CE%B9%CE%AC%CF%84%CE%BF.pdf?a9aeb9

From the centre of Patras to your hotel in Rio Achaia Beach
In order to arrive at the hotel by either the train, or the bus station you will have to take a taxi. Taxis in Patras and Rio are fairly easy to get. There is always one right outside every station. Should you have any difficulty you can hire one by calling +302610326000 or +302610 346700. The fair from Patras to Rio is approximately 8 Euros. Be aware that there is a double fare after 12:00 pm.