Hitler didn't have a beard. He had a moustache.

That said, the toothbrush moustache was popular well before his rise to power, and Charlie Chaplain was wearing one as early as 1914. It was particularly popular amongst German and Austrian men, owing to the fact that Hans Friedrich Wilhelm Hugo Koeppen - a German participant in the first world-touring car race in 1908 - wore one. There's no evidence that I'm familiar with suggesting that Hitler wore one to emulate Koeppen, but it doesn't seem unlikely that a young Austrian soldier might have been inspired by the heroic figure cut by the older officer.