No idea how old it is, BUT, if you can get a GOOD photograph showing the end-grain of the wood, you might have a methodology available. You could contrive to use "dendrochronology", which is basically counting the rings on a tree stump. Each ring in the tree counts one growing season... or year, generally speaking. Now, in this case, you cannot know when the tree that was used to make the end of the horn was cut down, so you can't count backwards (or foreward, for that matter) to arrive at a definite date... so this simple methodology would not work for you.
What is generally not known though, is that the rings of a tree are not all the same width. During good growing seasons, the rings are thicker than others. When combined, you have a pattern of growing seasons. That pattern is unique and defines a slice of time for a given area. Julian Barbour calls things like this "time capsules" in his book
The End of Time: The Next Revolution in Physics. Dr. Barbour preferred things like rock strata, but tree rings are sort of the same concept.
Since you have an idea of where the horn was made, all you really need is a tree that was cut down in that area that is maybe a hundred years old, or another tree that was cut a hundred years or so earlier (a known date). By this method, you should be able to arrive at a pattern that matches the end-grain of your horn. Even this won't give you an
exact date, but it will tell you approximately when the tree was still alive and growing. You can extrapolate the approximate age of the wood from that.
Say you find that the pattern matches the patterns of other trees during the years 1861 - 1865. This establishes that the horn cannot be older than that, though it could be younger, and because you don't have the entire pattern from the tree trunk, you cannot say for sure how much younger. If you can extrapolate the approximate diameter of the tree, you could potentially determine the approximate age of the wood at the point your particular piece of wood was taken from. If your particular piece was taken from a large diameter tree that was probably at or near maturity, you can figure that it was probably cut down about the time that the growth patterns match.
This would take a bit of work, and it won't give you an exact date, but it should get you within fifty years or so and it is much cheaper (and more accurate for this particular slice of time) than radio carbon dating.
Yeah, sometimes my inner nerd surfaces....
