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It sounds like your new kitten has an upper respiratory infection. Upper respiratory infections in cats are caused by viruses. The symptoms we generally see can include sneezing, discharge from the eyes & nose, coughing, fever, anorexia. Most of the time, these illnesses resolve on their own, but we often prescribe antibiotics to help prevent secondary infection in the nose, eyes and lungs.
This is not a medical emergency, we can support him through this if and until he needs to be seen by your regular vet. Symptoms can last anywhere from a few days to a few weeks.
Things that you can do to help: gently clean away any discharge from the nose and eyes.
Put him in the bathroom while you shower or in a room with a humidifier--just like in us, steam often helps clear stuffy noses.
To tempt him to eat, give her strong smelling foods like fish, and try warming the food up a little bit to enhance smell (cats will often not eat if they can't smell their food).
Keep him hydrated if he is not drinking--you can use an eye dropper or syringe to put water into his mouth, or if he will eat warmed up foods just add extra water to it (or tuna/chicken juice).
If the eyes have a pus-like discharge, this may indicate that eyes already have a secondary bacterial infection, which should be treated with a topical antibiotic ointment (available only from a vet).
I hope this helps, please let me know if I can answer any other questions.