I agree that with Donna that doing it as an excursion (either through the ship or some other venue Hotel, etc) would be better than trying to hire a taxi. It is approximately 1 to 1 1/2 hours from San Juan to El Yunque. That said it is well worth the trip. Absolutley beautiful.
We were on explorer this past march and booked an excursion to the rainforest through the ship. It was cancelled due to the fact we didn't have enough time. We got into san juan at 2:00. So be aware they will let you book it even though they "should" know it will be cancelled. We walked around old san juan instead and it was nice.