When we took the bus a number of times in October from Naxos town to various interior villages all the way to Apollonus we went over the mountains and while there were many switch backs and slow downs there was virutally no other traffic. You basically had the road to yourself.
As an aside we rented a car in Amorgos which also has high mountains, narrow roads and switchbacks in October and once again virtually no traffic. The only thing you had to keep an eye out for were goats.
While I've never driven in Ireland, let alone been there I doubt you'll see the type of traffic there as you'll experience in many Greek Islands especially ones like Naxos which while popular don't have the tourist levels of Santorini or Mykonos.
Take advantage of renting a car and really seeing a lovely Greek Island like Naxos.