I have not sailed from NO since Katrina, but assuming that nothing has changed here's the scoop. There is a parking lot adjacent to the terminal (on the dock). I've parked there several times and never had a problem. It is fenced and supposedly guarded 24 hours a day. There is an overflow lot also at a nearby hotel. It is covered parking and a shuttle bus runs between it and the pier every fifteen minutes or so. I've parked there also and had no problems.
As I recall, rates were about $7/day, but that could have changed. At one time you could pre-pay your parking through the cruise line but I do not know if that is still an option.
We sailed the Sun in November and had a wonderful time (even though it was from Galveston) and are scheduled to sail her again this coming November. It's good to have her back in Louisiana.
If you are staying in a hotel the night before you sail (and I hope you are after that loooooonnng drive ), they may offer free parking or it may be cheaper than the lot.
They usually offer a shuttle to the port or will call you a cab.
Just check with the hotel to see!
Have a great cruise on the Sun!
"A truly happy person is one who can enjoy the scenery on a detour."