This website, which appears to be from the Dodecanese Economic Development agency, includes tourist info in English, including a bus schedule. To get from Kolympia to Rhodes you would presumably take a bus run by the KTEL company from Kolymbia Beach, which runs
RHODES-KOLYMBIA BEACH: 09.00-12.00*-14.00*-14.30-15.00-16.00-17.30-19.30-21.15-23.00
KOLYMBIA BEACH-RHODES: 09.00-10.00-10.30-11.30-12.25*-13.30-15.00-16.00-16.25-18.15-20.10
PRICE: 3.90€ ONE WAY
(Starred routes do not run on Saturdays.) The timetable can also be found on KTEL's website.
You would then take the bus from Rhodes to the airport, using RODA, the Rhodes municipal bus company (can't find a webpage, sorry):
RHODES-PARADISSI(air port): 04.45-05.20*-06.30**-06.45**-07.10-07.30-08.15**-08.30-09.00-09.30-10:00-10:30-11:15**-11.30-
PARADISSI (airport)- RHODES: 06.40-07.25**-07.55***-08.15-09.00-09.20-09.55-10:15-10:45-11.15-12.10-12.30**-12:40-13:15-
PRICE: 2.50€ One Way FROM DRIVER: 2.60€ One way.
Here, a single star means the bus doesn't go down the coast beyond the airport; two stars mean no Saturday service; three stars mean only Saturday service. The above link also contains a Sunday/holiday schedule for this bus line.