If your cat starts sneezing and coughing and develops a discharge from the nose or eyes, he's probably suffering from an upper respiratory infection. Feline upper respiratory infections are viral in nature, so antibiotics won't work. However, it's quite common for a secondary bacterial infection to set in, and antibiotics do work on those. Take your pet to the vet for diagnosis and treatment. She'll decide whether his condition warrants antibiotic therapy.
A kitten's immature immune system means that a secondary bacterial infection is far more likely to occur if he develops upper respiratory issues. Your vet will culture samples from your pet's nose or throat to determine the cause of the infection. If the culprit turns out to be chlamydia or bordetella, she'll likely prescribe broad-spectrum antibiotics such as doxycycline or tetracycline to combat the bacteria. Make sure you follow your vet's dosing instructions to the letter.