From Sugar Land or Missouri City, budget 75 to 90 minutes down Beltway 8 to I-45 South and over the causeway. From Katy or Cinco Ranch, plan on about an hour and a half; from Richmond and Fulshear, the longer end of that. Coming from out of town, the Port of Galveston's own arrival guide puts Hobby (HOU) at roughly a 45-minute drive and George Bush Intercontinental (IAH) at about an hour and a half.
Everyone arriving from the north or west takes the same road — I-45 South onto Galveston Island, then Harborside Drive to the port. That is the pinch point. On a Sunday when several ships are turning around, the last few miles take far longer than the map says, so treat those numbers as free-flow and add to them.