I will be crusing from there in July, and was told that Hobby is closest. I have made reservations with a shuttle complany called Galvaston Limo Service (1-800-640-4826) It is 50.00 per person round trip and they have shuttles that leave every two hours.
Hobby Airport is the closest airport. Give yourself a minimum of 1 hr to get there safely from the airport. You take I-45 straight from Houston to Galveston. You can't get lost - one road in - one road out. I don't know what a cab would be - my guess too much. It is about 40+ miles from airport. Did your cruise come with transfers. I have seen in the airport people holding cruise signs.
I'm going in September but will be flying into IAH so I don't have to connect any flights and lose luggage. (Just me being careful.) Nick and I will be flying in a day early and renting a car in Houston and dropping it off in Galveston on the next day. Enterprise is in the process of creating a way for cruisers to drop off rental cars on Sunday even though they are closed. It should be ready in about a month they said. Are you going in a day early? I don't know where you are flying from but we decided that it was going to be to close to ly in on the Sunday. Airplane flights are actually chaeper for us on Saturday as well.