Hello, my name is***** and I have over 20 years of experience as a veterinarian. I’m sorry to hear that Mai Tai has red, watery, itchy eyes and a runny nose.
I suspect that he has picked up a contagious upper respiratory infection. An upper respiratory infection in cats is just like a cold in you and I, and these are commonly caused by a virus, the most common one being Herpes virus. These are easily caught by breathing in virus particles in the air from a sneeze or nasal or eye discharge. Some cats get sicker than others and young, healthy adult cats seem to tolerate them and fight them off better than very young kittens or older cats.
Some cats are chronically infected and will have reoccurring symptoms when they are stressed.
You can help him feel better by adding warm water to his food to make it smell more (they don't eat if they cannot smell) as well as making it easier to swallow.
Also the more fluids he gets the better. Offer tuna juice, low salt chicken broth, run the tap if he likes to drink out of the sink.
Take him into the bathroom with you if you run a hot bath or shower as the steam will soothe his sore throat and airways.
If his nose become very congested you can use sterile saline to loosen the thick mucous and remove it. He won't like it but it will help him breathe and be able to smell his food better. You can also use sterile saline to remove eye mucous if it accumulates.
To soothe his eyes you can flush his eyes several times with sterile saline. This is the same solution that soft contact lens wearers use in their eyes, ask the pharmacist or drug store clerk for help if you are unsure. A cool compress will help reduce swelling and inflammation around his eyes, and help them feel less itchy.
Then apply artificial tears every hour or two tonight. You can find these available over the counter at the drug store too. The more lubricated his eyes are the better for them and the more comfortable he will be.
You can give him an amino acid supplement called L-lysine at a dose of 500mg orally twice daily. If this infection is due to Herpes this amino acid interferes with virus replication and will shorten the infection's duration and severity. Good supplements to try are made by the Viralys brand which comes in a powder to add to the food or a tasty gel.
Some lethargy is understandable, let him rest as he needs rest to get better. If your cats normally go outdoors keep him (them) inside until they are back to their normal playful selves.
If you want to try Benadryl (diphenhydramine) to dry up his nose a normal dose is 1mg to 2mg per pound of body weight every 8 to 12 hours. So a cat his size could take one 25mg tablet every 8 to 12 hours. Some cats find Benadryl very bitter tasting and may drool or foam after the mouth after being dosed. This is strictly due to the taste and is not a true reaction. OR you could try another antihistamine, chlorpheniramine at 4mg per cat orally every 12 to 24 hours.
If he runs a high fever (more than 104F), has a green or yellow nasal or eye discharge, stops eating even with coaxing and clearing his nose and eyes, or starts coughing or having consistent difficulty breathing then he needs a veterinary exam. Not an emergency as long as he is eating but soon if he isn't eating or has a green or yellow nasal or eye discharge.
Sometimes these upper airway infections turn into pneumonia so that's what we need to guard against. In most cases antibiotics aren't needed and can contribute to a decrease in appetite so I don't tend to prescribe them unless I feel there is evidence of a secondary bacterial component. These include a green or yellow eye or nasal discharge, evidence of pneumonia upon listening to their lungs or an infection that lingers beyond the normal 7 to 10 days.
I highly recommend testing him for feline leukemia and feline immunodeficiency viruses if his respiratory infection lingers. These immunosuppressive viruses will make a simple infection much worse as they stop the immune system from fighting infections the way it was designed to do.
Let me know if you have any further questions.