It doesn't look to be as late as 13th Century to me. There are half a dozen types of helmets it could be, but my best guess is a spangenhelm, Central Asian in origin, but popular across both Europe and the Middle East between the 6th and 9th Centuries.

The crusades were mainly fought by professional knights and mercenaries, who could afford state-of-the-art armour (the enlisted peasantry couldn't afford to take the years away from the land it would have taken to fight an overseas war). By the 13th Century, this included full-face protection, such as a greathelm or bascinet.