It looks like a variant of the Eboshi. In kanji it would be written 烏帽子 and in hiragana as えぼし. (If you Google image search "弓道" and "えぼし" it brings up a few images similar to the specific type you had.)

That style of hats first appeared during the Heian Era (794-1185) which is remembered as the age of "classical" Japanese culture. Classical in the sense that it was a flowering of high culture from which modern Japanese culture is descendent. At the time Japan exchanged envoys with the Tang Dynasty and Chinese culture was all the rage with the Imperial Court at Kyoto. The Eboshi hat is an artifact of that Chinese influence.

Use of bow and arrow existed long before the Heian, but it began being practiced as an art, especially in a ritualized context during that era. If it is any indication, the word for archery in Japanese, "Kyudou," incorporates the character "弓" for bow and "道" for path, way, or more figuratively art. Its literal translation would be "way of the bow/ art of the bow".

Anyway, because of Kyudo's connection to the high culture of the Heian Era its traditional outfit includes the Eboshi hat. It is the remnant of an ossified tradition. Even today when Kyuodo is preformed during festivals and celebrations the ceremonial getup includes Eboshi hats.

I hope this quick summary helps or at least gives you a jumping-off point.

*Edited for formatting