Agreed with Rebecca, you can't pass a cold to your cat (you can pass on certain eye infections and other stuff but that's another topic). However, if your cat does have a cold it needs to be treated by a vet as what is commonly called a cold among felines can get very, very bad. Cats can lose their eyes for example, but those are of course cases that have been left untreated for a long time, so don't panic either :)
If the cat regularly sneezes, or has more eye discharge than usual (especially with red eyes or the pink skin underneath the eyelid (conjunctiva) easily showing), or has trouble breathing, or has, well, snot running out of their nose... Definitely time for the vet.
They will usually prescribe antibiotics that should be given until a day or so after all the symptoms have passed, so get more if necessary.