It is likely that your kitty has picked up an upper respiratory infection. Because he is holding one eye closed, I would be most suspicious of infection by a Herpes virus. This virus can also cause ulcers on the surface of the eye and be quite painful. The fact that he is holding the eye closed means that the eye is painful.
There are several viruses besides Herpes that can cause upper respiratory infections in cats, and antibiotics are not successful in treating these, but there are also organisms that can cause these symptoms that will respond to antibiotics (Chlamydia and Mycoplasma).
It is time to have him checked by your vet. If he was just sneezing but was feeling good and didn't have the eye problem, it would be okay to wait and see, but since he is lethargic and he does have some eye involvement, it is safer to have him seen. Eye problems can worsen quickly and sometimes how quickly they are treated will determine whether the eye retains vision.
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Herpes virus is an infection that cats get that can cause the symptoms you describe. They get it from other cats who have the infection, but it very contagious and can be carried through the air, so it does not necessarily require close contact between two cats. Your kitty sitting in an open window could allow exposure to the virus from a cat that is outside.
It would be extremely unlikely that any upper respiratory infection would be fatal unless there was something else going on (such as feline leukemia or feline immunodeficiency virus) so taking him in tomorrow should be just fine. The exception to that would be if you are noticing any difficulty breathing (not just stuffy nose but rather a change in his breathing pattern). That could indicate a much more serious problem, but since you said you are noticing sneezing rather than coughing or difficulty breathing, I am assuming these things are not present.
Please let me know if you have additional questions.