Is the amoxicillin liquid or pill form? If it is liquid and constituted (mixed with water) it should be discarded after 14 days.
Were the cats about the same weight? You may be able to use the same dosage. But, either way, I don't like to use the same prescription provided for one cat for another, due to dosage requirements per pound, unless you're educated on this. Just as in Humans, you should never take someone else's medicine.
Sounds like your cat has developed an upper respiratory infection. Or...has your cat had it's shots up to date. Has your cat ever been tested for Feline Aids? If the answer to any of those questions is no, then take the cat to a Vet.
Also, you can use an immune enhancer called L-Lysine to sprinkler into the cats food that will help to boost the immune system. You can get this as an over the counter drug at your local Wal Mart in the Pharmacy next to the Vitamins. It will come in pill form and you'll need a pill crusher. I give my geriatric cats 1/4 tab a day.
NOTE: This is not a replacement for going to the Vet. This is a supplement for when the cat is Healthy once seen by the Vet and given proper medical care.
Adamant volunteer for our local Animal Foundation, and advocate for the T-N-R (trap,neuter,release) programs to bring down the numbers of over populated feral cat colonies. Licensed foster care provider for homeless cats/kittens, incorporating Behavior Modification to rehabilitate adoptable ferals and find them forever homes.