Maple Shortbow

Archer's Arm, item level 5. Equip level 5, ARC BRD.

From 20 Passing Muster and 1 more quest.

From quests

20 Passing MusterChoose one of 4
21 A Matter of PerspectiveReward

Other databases

Eorzea Database (the official one, on The Lodestone)
Console Games Wiki
Garland Tools (vendors, crafting, drops)