The universe appears "old", but it is still less than 10,000 years old

[popeyespappy]

Looking at the night sky is like looking at the lights on Earth at night. While you can calculate the distances between the lights it's an error to think that that distance means that the objects are a certain age. In other words, distances does not translate into age. We are supposed to have local star clusters that are almost as old as the alleged age of the universe.

A more reasonable statement might be that if you could travel at the speed of light and didn't hit anything it would take you 13 billion+ years to reach the farthest object you saw 13 billion years ago. By then you would have a whole series of new problems.

Distance does translate indirectly into age. It doesn't tell us the actual age, but gives us a minimum age that object can be.

Yes, if you traveled light speed it would take you ~13by to reach the farthest object you saw, but it also means that if we can see it now, it took its light ~13by to reach earth.
If the object were 13b light years away, but was only 1b years old, we wouldn't see it for another 12b years.

It is actually a little more complicated than that. If you are looking at light that was emitted 13 billion years ago you are seeing the object as it was 13 billion years ago. However the object was closer than 13 billion light years to us when the light was emitted and is now further than 13 billion years from us. This is due to the expansion of the universe.

There is a short article that talks about this effect here. According to the article the light we received from the most distant object ever observed took a little over 13 billion years to get here. The distance to that object is now about 30 billion light years due to expansion.

There is a calculator here that will give you both the distance when the light was emitted and the current distance if you have enough information.

According to this Wiki article we should never be able see any objects that are currently more than 62 billion light years from us because the distance between here and there is expanding faster than the speed of light.