that sneezing and runny nose stuff and eye gunk are NORMAL in shelter kitties. They are all symptoms of Upper Resperatory Infections, which are to shelter cats what chicken pox are to daycare kids. Even if the cat you pick shows no symptoms, the stress of leaving the shelter and coming home with you may trigger one.
It is a simple infection, and can be cleared up with antibiotics.
However, they are spread VERY easily, so as Imagyne says, keep the new one separate from the others for two weeks. The cost of treating all your cats with antibiotics will be very expensive.
Also, ask the shelter if they test for Feline Leukemia and FIV. It is just good to know. If they don't, take the cat to the vet immediately for a test. These can be spread to your cats too, and there is no cure.