While beech is classified as a hardwood, it is not all that "hard."  While it is denser, it is not that much denser than pine, certainly much  less "hard" than oak, but beech does have a very uniform grain that makes it easy to work.  It doesn't split out as easily as pine.
 
Also, be very careful of the "all wood" moniker.  Press board covered with a veneer can be classified as "all wood" and press board falls apart when it gets wet.  Take a close look at the back side of the case.  If you can find a true edge grain on the back side then it is probably not press board as they don't go the the effort to conceal those edges.
 
Whether pine or beech, it will need a good coat of paint or marine grade varnish to protect it from both the weather and UV radiation from the sun.
 
Good Luck!