Should I take my kitten to the vet?

If it were me I would probably keep an eye on him for a few days and see how he does. If there's no improvement, I would call the vet, explain, and ask if you can use the same medication. It's possible they would want to use a different medication this time so that's why it's good to ask. The other thing that can happen is there are a few viruses that can cause the upper respiratory infection. Some are viruses that will stick with a cat for a lifetime. Then they can flare up during times of stress. If this is the case, something like lysine could be more helpful than antibiotics.