Despite Fry, being frozen, and remaining the same age for a thousand years, if someone was to carbon date him, or by birth certificate wouldn't he be 1,02X years old?
To answer this, it's important to understand how carbon dating works.
Basically, when plants photosynthesize, a certain percentage of atoms they get in CO2 from atmosphere are unstable C14 isotope. They get incorporated in the plant tissues, and the percentage of C14 there gets roughly the same as in the atmosphere. You eat the plants or you eat the animals who ate the plants, so your C14 ratio is also close (some notable exceptions, like seafood, apply).
Once a plant or animal dies, it no longer lives (duh), so no new carbon atoms join it. It does decay, but this affects molecules, not atoms. So, stable carbon isotopes remain while unstable decay. Ratio of C14 compared to stable C12 gives the estimate of time our specimen spent dead and buried.
We may assume that cryosleep is similar to death in this aspect: no new atoms entered Fry's body during all that period. So when he woke up, he would be carbon-dated as a 1000 year old fossil. But as soon as he woke up, he began to eat, drink and breathe, gradually bringing him closer to other people in terms of dating. But not completely. If this answer is correct, Fry's teeth will give away his chronological age.
PS: Birth certificate? Yeah, 1000+ years. So what? There's a freaking alien on news channel, these people won't be shocked by an old young man.