Polymyxin B Sulfate and Trimethoprim Ophthalmic Solution ..can this be used to treat a stye?

Not unless you were prescribed this medication by your veterinarian for your cat. Human formulations of this combination antibiotic are dosed for humans; they are often too strong to use in cats and cannot be safely diluted for use in cats. Also, in general, cats get viral infections in and around their eyes, not bacterial, so antibiotics would be useless anyway.

If your cat is having issues with its eyes, the best course of action is to take your cat to a veterinarian. After diagnosing the issue, your veterinarian can then prescribe medications if that is what your cat needs. Trying to resolve the issue at home with medications not intended for use on cats can cause your cat to lose his eyesight, or even his eye.