We have been on this road many times (we have stayed along the Amalfi Coast) and often drive it in our own rental car. Yes, there are a lot of curves and some of the view is spectacular with curves around the top of seaside cliffs. But, its actually a very safe road with strong stone walls and guardrails protecting the dangerous edges. Positano is one of the most picturesque towns on earth, so its one of those places you should see at least once.
I want to be careful on this answer. Yes, we rent cars independent of the ship, and we do it in lots of European ports. However, we have also been doing international travel for over 40 years and much of that has been on our own on both land and sea. In other words, we are very comfortable driving in Italy and the rest of Europe. If you are spending a typical port day in Naples, and have little or no experience driving in Italy, than I would not recommend you rent your own car. Driving in Italy is fun, but its also a little crazy, and when you get onto the Amalfi Drive (assuming you do not get lost and can find the road) its entirely 2 lanes with a lot of traffic. If you want to do this on your own than you need to get going early in the day, drive to your furthest point (probably Amalfi or Positano) and work your way back so there is little danger of missing the ship. You also need a "plan B" in case you would miss the ship,,, which means how you catch-up the next day.
Just back from a Celebrity cruise to the Med. Did a private tour of the Amalfi coast with an independent tour operator at a fraction of the ship excursion prices and without the headaches of driving ourselves. 8 of us travelled in a comfortable air conditioned van from Naples port (8am) to Sorrento, Amalfi, Positano & Ravello...back to ship by 5:30pm. Alberto our english speaking driver was very accommodating and personable. I would highly recommend taking this tour. It definitely was a highlight of our trip. See www.gianpierofiorentino.com