Coughing Kitty: is my cat seriously sick?

He probably has an upper respiratory infection. They move through shelters and stray populations pretty quickly. You can watch him for a few days or a week and see if he gets better. If he doesn't get better, you should check in with your vet. Many times these are viral infections they will get over on their own. Sometimes they get a secondary bacterial infection that would benefit from antibiotics. Once in awhile it's something entirely different but URI is most likely. Thanks for adopting :) Good luck!