Is it theoretically possible to determine the age of the universe to a high precision? My question is a bit abstract, but I'm wondering if it makes sense to expect/hope that one day we will be able to measure the age of the universe (in Earth's frame of reference) much more precisely, down to sub-year precision? Is that a well defined question and is it physically possible to figure out?
I'm thinking that maybe the expansion of the universe or something to do with relativity could make this impossible to answer or a nonsensical question to  ask.
 A: Within our present model of the universe, the age of the universe is deduced from observational data :

Ancient light from the Big Bang has revealed a precise new estimate for the universe's age: 13.77 billion years, give or take 40 million years.

13.770.000.000  earth years +/- 40.000.000  earth years.
If we accept the claim that it is the best precision, (ignoring maybe larger errors due to models used),the  error given is   too far removed for your "sub year precision" for it to be  attainable, imo.
A: Even assuming we have the correct cosmological model, the precision to which we can measure cosmological parameters is fundamentally limited by our vantage point. Our location in the universe is random, but every experiment we do is from the same random location. We can reduce instrumental error in measuring the cosmic microwave background, but we are always measuring the same inherently noisy pattern. This problem is called "cosmic variance".
In principle, we could discover a theory of everything that forces a certain value for certain cosmological parameters (perhaps the cosmological constant), allowing us to better fit the other parameters, or we could find maps of the microwave background from millions of years ago in the ruins of Atlantis and combine it with the current map to reduce the noise. Failing that, I think that future astronomical observations can only slightly improve the precision of the fit.
A: The key concept in the question is. "Is it theoretically possible...?" This a distinctly different question than whether it is practically possible. As astronomical instruments get bigger, their precision will also get better. The biggest current instrument in recent times is the Hubble with a diameter of 2.4m,  which has produced an error of about 40,000,000 years. One can imagine theoretically creating a telescope the size of the solar system with a diameter of about 22,000,000,000,000m. If such an instrument were to someday be created, it would not be too surprising to me that the precision of the age of the universe will very possibly become less than a year.
