We would like to take a ferry from Santorini to Naxos and take a taxi to Plaka Beach, Naxos. Can we accomplish this in one day? It looks like the earliest ferry from Santorini to Naxos arrives at noon. The return ferry leaves at 4:00 pm. Is it possible to travel from the port (not sure which one) to Plaka Beach via taxi within that timeframe and still leave a few hours for the beach?
Plaka Beach is 7km from the ferry dock. It won't take more than 20 minutes to get there by taxi. According to your stated arrival and departure times that would safely give you 2.5 hours at the beach.
Since you didn't give a date for this trip I was unable to find the ferry schedule so I will assume that you have that information correct.
The main taxi stand in Naxos is right at the ferry port. Just walk off the dock and go straight and you will see several silver taxis.
It will be about 15 minutes to Plaka beach. 15 euros I think
Getting back will require your attention. You can ask your driver to be back at Plaka beach at a specific place at a specific time. I might suggest 3:15. 3 would be better if you want to take a stroll around Naxos town. Or you could get the drivers cell phone number and call for the taxi when you want it. Of course that way its 15 minutes out and 15 back.
There is excellent bus service from the port to Plaka beach. Its every hour in shoulder season and every half hour in busy season. Probably not practical for you since you can't afford to waste any time.
Here is what you can expect at Plaka. Note the further out you go the more skimpy the bathing suits become
Plaka beach tavernas hotels https://www.flickr.com/photos/stanbr54/sets/72157655106193251
Visiting Plaka and getting back to Santorini on the same day is feasible but not very convenient. The earliest ferry from Santorini to Naxos arrives at 12:10 and the latest departs from Naxos at 16:10 as you can see in this picture (source ferryhopper)
Finding a taxi is rather easy (they are parked at the port, just outside the ferry) and arranging with the taxi driver to come and pick you up from the beach is also easy, but still you still you 'll just have 2 hours to enjoy Plaka (your time will be limited between 13:00 and 15:20 given the fact that there will be some traffic both in Plaka and in Naxos town)
Indeed Plaka is a place that you would like a lot. So, I don't know if it would be possible for you to stay in Naxos overnight, but if so, you should already start looking for a room (Plaka has plenty of hotel/room options). Then, you will have enough time to enjoy the beach, an fantastic lunch in Cedar Cafe and an amazing dinner in Petrino :)
ps. ...and then, you will forget about Santorini!