1. Other states following South Carolina out of the Union is most unlikely. First, it's important to remember that Old Hickory, a Tennessean and strident Unionist, was President and head of the Democratic Party.In 1832, South Carolina's John C. Calhoun had not yet attracted the broad, regional following for his "southern slave power" politics that would destroy the national Democratic Party, not even close to it.
It took decades for Calhoun's philosophy to poison southern minds and turn them against the Union. The only state existing at that time that might -- might -- have followed SC out is MS, and given how lonely a stand that would have been, it is very, very implausible. That much is very self-evident in the facts of the political history of the period. To assume otherwise, you have to rewrite a whole lot more than just saying South Carolina jumps off the plank in 1832.
2. So, we have only SC leaving the Union. This is 1832, so it is before the existence of the railroad, but South Carolina is a coastal state with some big west-to-east rivers in it. Logistics would not have been a problem, and the Palmetto State would have faced the might of the entire United States under the iron-fisted leadership of Andrew Jackson. They would have been completely, utterly crushed.
3. As some others have pointed out, settling the secession question with Jackson crushing rebellion in South Carolina in 1832 would have put paid not just to the legal idea, but also to the political philosophy that gave rise to it. As I noted, it took decades for Calhoun to work his witchcraft on southern minds and turn a bare majority of them towards hardline sectionalism. South Carolina might have continued to burn with the resentment of the loser after that, but Calhoun's political agenda would have been consigned to the dustbin.