While it might be possible for you to find someone who is good at fixing altimeter watches to do this off-warranty, my expectation is that you still wouldn't be able to get this fixed. If the backlight has failed, the chances are that you won't be able to obtain the parts.

If I were you, I would look for a replacement or carry a small light suitable for checking the face of your altimeter watch. Casio has altimeter watches starting at under $50, which is certainly less than any attempt to repair it would cost.