distances - Is the light we see from stars extremely old?

Yes, the speed of light in vacuum (or c) is 299,792,458 m/s and one light-year is the distance the light travels in one Julian year (365.25 days), which comes out as 9.4605284 × 10¹⁵ meters. Since c is the maximum speed at which all energy, matter, and information in the Universe can travel, it is the universal physical constant on which the light-year (ly) as one of the astronomical units of length is based.

That means that visible light as an electromagnetic radiation cannot travel faster than c and in one Julian year it can traverse a maximum distance of

d = t * c

d is distance in meters

t time in seconds

c the speed of light in vacuum in meters per second

If we calculate this distance for a 4.243 ly distant object, that comes out as 4.243 * 365.25 * 86,400 s * 299,792,458 m * sˉ¹ or exactly 40,141,879,395,160,334.4 meters (roughly 40 trillion kilometers or 25 trillion miles).

That is the distance the light traveled since it was last reflected of (or in our case emitted from, since Proxima Centauri is a red dwarf star) the surface of a celestial object to be 4.243 Julian years later visible at our observation point, in this case our planet Earth from where the distance to Proxima Centauri you quoted was measured.

The more powerful the telescope, the further into the past we can see because the light is much older! This goes the same regardless of the distance of the object you're observing, but astronomy is particularly neat in this regard and we can observe objects that are so distant that we see them from the time when they were still forming.

For further reading on other units used to measure far away objects you might be interested in reading this question on the parsec.