Is my Zebra Finch sick?

If it has only been about an hour, you should probably wait a day or so and then if it continues take him to a vet. Unless there is something more serious, I would not worry about it just yet.

It's normal for zebra finches to take a rest and fluff up for for an hour or so. Keep an eye on your finch for the next few days and see if he remains that way. Finches and other pet birds fluff up to conserve body heat. If he remains fluffed up all the time, then take him to see a vet because that is a sign of illness in birds. Other signs of illness in finches are: sleeping alot, sneezing, loss of feathers, frequent itching, laying on the bottom of the cage, not eating, vomiting, diarrhea, dirty feathers, bird doesn't sing.