I had read several threads where people were saying the garage for cruise passengers was the Aquarium parking garage. I was just reading a thread on Cruisecritic.com about this. The garage is called Republic Parking Garage and is located at 24 Calhoun Street and is $10/day. I don't know Charleston very well, but from the maps I've seen on the internet, the Republic garage is located very close to the aquarium, but is not the Aquarium garage. So hopefully the difference won't cause problems for passengers.
Here's the link to the discussion, you may have to paste this into the address bar.
I have no idea since I haven't seen it, but if you post your question on the cruisecritic message board link I provided above, the person who visited the port area, may be able to provide a specific answer.