What terminal does your flight arrive at? If it's Terminal B, the main terminal, you'd have to take a shuttle bus to the pickup point. So for that terminal a taxi will definitely be better.
What time Tuesday do you arrive? Are you arriving the same day as your cruise departs? That's always somewhat risky.
Shared van airport shuttles are all terrible in NYC. There are so many other transportation options and I think that's part of the reason why the shared van shuttles are so awful.
Most people don't go straight from the airport to the cruise terminal so I doubt there is enough demand for that route to have a dedicated shuttle. What if your flight is late or canceled? Many people want to come in the day before and stay overnight.
If you decide to risk flying the same day as your cruise departs, I would just take a taxi to the cruise terminal. Go to the official taxi stand outside your airport terminal. Ignore anyone approaching you offering some other type of taxi or car. Those rides are illegal and people have had terrible experiences of being intimidated into overpaying after unknowingly accepting the ride from a convincing tout.
I'd say $40. It's only 11 miles or so. And the worst traffic part is getting from 2nd Avenue to 12th Avenue, i.e. those last 2 miles. The first 9 miles should not be too bad. But I assume you don't have to be there before 4 PM, and even if your plane is 4 hours late, you will be fine.
Can you drop your luggage at the ship or do you need a place to store it?