Yes tobramycin is safe to use in a cat's eye as long as it is an ophthalmic formulation. If you do not know if the cornea is intact do not use any opthalmic medications with a steroid component.
If the eye does not improve after 1-2 days of treatment have your cat examined by a veterinarian. Ocular foreign bodies cause conjunctivitis that does not resolve with treatment.
Yes tobramycin ophthalmic drops are used in veterinary medicine to treat conjunctivitis in cats. This is an off-label use however--there is no veterinary approved and licensed product. It NEEDS to be an ophthalmic preparation--specifically formulated for use in the eye. Tobramycin can be used in the affected eye 2-4 times a day. Tobramycin is not the right antibiotic choice for all infections.
I would not use any medication without having your cat examined by a veterinarian to make sure you are treating the right condition.
I would buy something that is specially made for your cats eyes. Call your local vet or pet store, like petsmart, and buy something there. At least check and make sure these drops are safe. The vet would be your best vet. Make sure they are for the cats eyes too, like previously stated. I am not aware of these drops. Cats have different vision and see different then us, so you don't want to hurt the cats eyes. Usually, if they are goopy, you can get a wet wash cloth and wipe them and clean them a few times a day is best. Then if they are infected get eye antibiotic drops from the vet, they are not too expensive and use those, that are made for the cats eyes, is what I would do. Good luck.
No, I would recommend you use a warm tea bag to help you with his eye.