I was on Naxos way back in Oct. 2013 and it was lovely. Pleasantly busy but without a lot of tourists.
I relied on the bus to take me around the island and was satisfied with it.
However, I soon realized the bus doesn't go everywhere, nor on my schedule. Let alone stop for the numerous photo ops or linger longer in a village or archeological/historic site and hope to catch the return bus.
It's a matter of budget & worry about driving in Greece (mainland/island).
Once I started renting a car I found out it's the only way to go for me anyway. I can go where I want, when I want, take a back road and find out there's more to Greece than just where the bus goes.
Still if you feel more comfortable sitting back and looking out a bus window with the bus driver doing the work then you should be able to see a lot of Naxos.
You don't have to rent a car for your entire stay but a couple days will give you the opportunity to find those out of the way areas of Greece tourists don't go to.