Did anything in his environment change? Like you using different cleaning products to wipe the floor or other surfaces or you using a scented candle or a room scent dispenser? If so, stop that immediately and air the room very thoroughly. Cats are very vulnerable to irritations of the airways by essential oils and many essential oils that are harmless to humans are actually poisonous to cats. You can read more in this question and this one.
If there's no irritant in the air, he may have any of a host of illnesses, like a simple feline cold or upper respiratory infection, heartworms, asthma, lymphoma, a fungal infection or even congestive heart failure. Make sure you get him to the vet on Monday because only a vet can diagnose and treat the real cause.
If you're worried about his condition, gently lift his lips and look at the gums. In a healthy cat they should be pink and wet, just like a humans gums. They can become pale and sometimes with pale yellow/beige tint if the cat is ill. If the gums are tacky or sticky, your cat is dehydrated. An ash-grey or blueish tint indicates lack of oxygen and you should contact an emergency vet ASAP.
Do not give your cat any human medication unless instructed by your vet! Please read this question for more information.
In the mean time, you can try to alleviate his cough with home remedies:
- This site suggest bringing your cat into a warm and steamy bathroom for a few minutes. The steam is supposed to open the airways and fight the infection. A room humidifier can achieve the same result.
- Wrap a hot water bottle in a towel and offer your cat a warm place to cuddle up to. An electrical heating pad offers the same comfort.