Kosta , the mainland port for Spetses, is only a very short boat ride ; passenger ferries and taxi boats. No private cars are allowed on the island.No doubt a local travel agent would put you in touch with a car hire company, but it would be very expensive to return the car to a different town, than from where you started the journey.
Maybe take a Flying Dolphin from Spetses to Portoheli and catch a bus to Nafplio, a good centre for exploring the ancient sites , car hire available , and a frequent bus service to Athens, if that's what you need at the end of your trip
Just had a look at the bus timetable on www.ktel-argolidas.gr; this shows two buses a day between Nafplio and Kostas, doesn't show the return times, obviously they exit. So it would be a matter of discovery when you're on Spetses. Or perhaps a resident or someone who has used the service can help?