It sounds like he has an upper respiratory infection, as these are common in cats, especially cats that are outside but as Dr. Amanda said, the best thing to do would be to have him evaluated by a veterinarian as there are other things it can be. If that is not possible, however I have put together some information for you that I hope is helpful.
Most upper respiratory infections in cats are viral so antibiotics are not necessary. Antibiotics are only helpful if there is a secondary bacterial infection. Even so, most cats will be able to fight this off themselves with proper supportive care. In severe cases, antiviral medications can be helpful and may shorten the course or decrease the severity of the infection (the one I use most often is called Famcyclovir) but this is available by prescription only so you will need to have the cat examined by a vet in order to get a prescription. The medication itself is not usually cost prohibitive and I usually use ½-1 250mg tablet twice daily for 14 days. This can be give for longer if necessary. You can have it compounded by a compounding pharmacy into a flavored liquid that he may accept if mixed with food.
While some people advocate the use of L-lysine for cats with upper respiratory infections a recent study did not show this to be effective in reducing the duration or number of outbreaks, so I do not feel strongly about using it. It will not hurt but it may not help. There are many supplements available over the counter if you want to try them. Most are flavored to be more palatable to cats so he may accept this if mixed with a yummy canned food.
Speaking of food, you can encourage him to eat by heating up canned food as cats with upper respiratory infections often do not eat because they can't smell very well. You can also try baby food (meat based with no onions or garlic), tuna or chicken juice to encourage him to take in more fluid.
If he will let you handle him, you can clean the eyes with warn water and a soft cloth. If there is nasal discharge you can use saline drops to break up the mucus and help him or her breathe better and you can put him in a steamy bathroom or use a nebulizer to help clear discharge as well. Of course this is all dependent on how much he will let you handle him, there are many limitations to being able to help him if he will not let you pick him up or he cannot be brought inside.
You can use regular saline eye irrigation (or contact lens solution) to clean around the eyes as well as this will not hurt if it gets in the eye. As upper respiratory infections in cats can cause corneal ulceration, there is an antibiotic eye ointment that is available over the counter in some pet stores or feed stores called Terramycin. This can be used in the eyes 3-4 times a day (or as often as he will allow you to apply it) to help prevent secondary bacterial infection of the conjunctiva.
Since catching him is going to be a challenge, I recommend going very slow with him to show him that you will not hurt him. Petting him is a good start and if you take things one at a time with him, he may allow you to do more once he becomes comfortable. If you cannot wait to tame him slowly because he needs to be treated and examined by a vet, you can use a humane trap (usually available for rent from your local humane society or feral cat coalition or can be purchased online).
Although it sounds like an upper respiratory infection, its possible that the clinical signs you are describing have a different cause- allergies, parasites, pneumonia, among others.
I hope this answers your question. If you have additional questions or want to discuss your cat's specific clinical signs in more detail please let me know.