Unless SC has weird rules like KY does, they will probably just hold the special election at the same time as the general, with the same candidates, since the general election is so close. They might be able to consolidate the two elections into the same ballot line, but maybe they can't, and there will be two lines the ballot for the same job.

So someone will be appointed to the job soon. Then, that person will get the job until whoever wins the special election is certified, probably mid-november. Then, whoever wins the General Election will take over whe the term officially begins in January.

Since SC is a safe Republican seat, though, the current Republican nominee will probably get appointed to the seat, and win both elections, so he will keep it until 2032.

Edited to add: ah, it looks like Graham was already the nominee. This will put SC Republicans in a bit of a time crunch to select a replacement. Because of that, it's possible they appoint a caretaker to only take the job until November, so that they can take the proper time to find a replacement on the ballot....

In any case, there is zero chance a Democrat can get this seat in SC. Even if Republicans can't pick a replacement in time for the election, SC would still elect a dead Republican over any Democrat.