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How old is he?
Is he breathing labored when he is not coughing?
Any congestion in nose or eye discharge?
A cough can be an indicator of few different things.
It sounds s though your kitty as an upper respiratory infection (URI).
What you can do at home to help him feel better is keep the nose and eyes clean of discharge. You can use warm water dampened cotton balls or Qtips to do this. You can also bring him in the bathroom when you shower. This will allow the steam to help break any congestion in the head and chest. A vaporizer can well too for this. It is very common for them to stop eating when they get a URI. Often times they can not smell as good, and cats don;t eat what they can smell. Try using odorous foods and warming them in the microwave. Just a few seconds and stir really good. You don't want hot spots.
You will need to get an antibiotic from your vet to clear it properly. Sometimes if URIs are left untreated they can cause eye ulcers or even pneumonia.
Right now it is an easily treatable condition.
Please let me know if you need further assistance with this.