Yes, the port is up and running. However, the rest of the island is still recovering. Hobby is the closest airport, however, International has many more flight options. Southwest flies into Hobby.
It is costing me as much to cover the distance from International to the port as it is to fly over 1000 miles to get to International. The problem is, even before the hurricane there were no rental car offices on the island. Thus, you are looking at cruise line transfers, taxis or limos. I got a private Town Car for the same price as a taxi would cost me. On the way back I'm taking one of the cruise line's shore excursions, which includes a tour and then finally being dropped off at the airport.