You might want to call or email your hotel in Sorrento. They might have a car service that they work with which might have a discounted rate. For example, we have stayed at the Hilton in Sorrento. Not only will they arrange for a pick up at the airport, they will even allow you to charge the fare to your hotel bill. The advantage to that is - you can get "points" on your credit card.
The other benefit is that because it's the hotels recommendation it will be a reliable service and they will track your flight in case it doesn't arrive on time.
Also, the rate is generally charged in euros, so the rate you posted in dollars seems high. By the way, we had an airconditioned Mercedes. It was well worth it. In our case, the driver met us inside the airport, holding a sign along with all the other drivers waiting for their passangers. We only had carry on bags, so by the time we entered the airport we were at the hotel in less than 90 minutes.