Actual boarding port is adequate for the process. We were the only ship sailing from the port that day, but the process could have been better. Mainly due to standing outside (covered) when there was a very large, cool (empty!) area inside seat up to handle the crowd.
I am going to answer this as if you are looking at arriving a day early and wonder what it will be like.
I like San Juan as an Embarkation / disembarkation port. It is located on the north side of the island on a harbor, with beaches facing the Gulf of Mexico. It makes a wonderful backdrop for Sail away and return. This is not just a sleepy Caribbean town.
The airport is modern, being updated even in Spring 2023. It was very busy the day we left, but could handle the crowds.
The original city of Old San Juan (OSJ) is about 30 minutes from the airport, but you may opt for a hotel near the airport, near the cruise port, or halfway between airport and OSJ. It is the second oldest towns in the Western Hemisphere (1519). The buildings are colorful and beautiful. There are old forts, streets built with blue bricks that were used as ballast on ships that sailed from Europe, and cats that wander around (keeps down any rat problem). It can be walked in a day. Lots of food and beverage options.
Less than an hour west from the airport is the El Yunque Rain Forest. Rent a car to visit or take a tour.
The cruise port is across the water from OSJ - a walk would not be done by a typical tourist with luggage. For ships visiting as a day stop, there are piers right at OSJ.