Yes, cats do indeed get upper respiratory infections, which are commonly referred to as colds. Most of the time, when cats get these types of infections, they sneeze a lot and may have some watery eye discharge, but they usually feel good, continue to eat and drink and act normally. The fact that your kitty is lethargic and does not want to eat, makes me more concerned. Because of this, I would definitely advise having him checked out by your vet.
In the mean time, it is very important that we try to keep him eating as cats that go very long without eating are prone to develop a liver problem called hepatic lipidosis, which is very serious. If you have not already done so, you may offer different types of canned food, boiled chicken or even tuna. Sometimes warming the food up in the microwave can make it more attractive to cats. If there is any discharge around his nose, be sure to keep this area clean as well.
Please let me know if you have any additional questions.