I would say he's probably 3-4 weeks old from the looks of him.
If he isn't eating solids--not even kitty mash (dry food moistened with warm water) or canned food for kittens--then go out and get a milk replacement formula for cats (KMR is a good brand, but I know even the local Wal-Mart in our area carried a kitten milk replacement) and a bottle. Now, I have mixed feelings with the bottle because it has never worked for me.
My girls came to me when they were about 1 week old and never took to the bottle. Galina was so weak she couldn't suckle anyway. So, if you find he's not taking to the bottle (and give it a day or so to be sure--he's not going to like it at first, anyway, because it is not mama's teat and doesn't smell like mama), go out and buy a large syringe, preferably one with a tapered tip to make it easy to insert into the mouth.
Next, at his age he's probably just starting to go potty on his own, so you want to buy him a small litterbox (or even a plastic bin of some sort that you cut a hole into the side) and a NON-CLUMPING litter. The non-clumping factor is very important since little kittens sometimes play in their litter and at first, might try to taste it. If the litter is the clumping variety, it will clump up in their system and cannot be passed out--very disatrous.
Gerona started pottying by herself when she was 3 weeks old and never looked back. Maybe it was because 3 days after I received her, she had to have an enema and since then has wanted me NO WHERE near her hiney.
Galina, on the other hand, was still needing a little help at 4.5 weeks but by 5 weeks was going on her own. Just monitor it--if you don't see any signs of pee or a poo in 1-1.5 days, try stimulating him as Jade mentioned above. If by 2 days you still get nothing, call the vet because he may be constipated and might have to have an enema. By three days, it's serious and you need to see what's clogging him up.
And lastly, he might not have an appetite because he's sick. Gunky eyes is a common symptom of an Upper Respiritory Infection (URI) and so you will want to take him to the vet so they can check him out.
Galina was 5 weeks when she came down with a URI and her appetite dropped IMMEDIATELY. But, after about 2 days of medication, her appetite came back and has been normal since then.
Keep us updated on his progress!