I live near the Port of Houston. There are shuttles from IAH. You can call the cruise line and they will give your information and I believe they even offer travel vouchers...
If you are driving you should go 59 not sure of direction, but when you get to 59 go to the right to Beltway 8 South - sorry readytogonow, this is much closer. Take the beltway to 225 east and continue east toward La Porte. You will see signs for the terminal. Nevertheless, it is the Barbours Cut exit. The port is not beautiful as it is a cargo ship port.
Renee