Well, as that act is subsequent to the 1805 act which granted Sumter to the US, it would appear to supersede the 1805 act. This act of 1836 also contains some conditions, but I will have to read it more closely to determine if these are dispositive or not. It seems that the conditions "That all processes, civil and criminal issued under the authority of this State" applying at the Fort, might be a loophole.
Actually no. What that means basically is that the local laws, criminal and civil, apply on the grounds of the fort. If a soldier there robbed a Charleston merchant then they could not take refuge in the fort free from arrest. If a Charleston civilian filed suit against someone in the fort then the local authorities could serve papers. Basically any property that a state cedes to the federal government has a clause like that in the legislation that cedes it.