Just what it says on the tin.
Bows and Arrows
Whereas I am not terribly for the idea (thought not against it), the bow could be held in the shield hand just fine. That arm stays rather still, anyway.
A crossbow would make more sense, visually and logically (not like logic has place in this game), though.
I like the visual concept of non-gun ranged weapons. They'd behave basically the same as guns, just with different looking forms and bolts- think Slime Slasher shovel.
This might be especially interesting if bolts can be coded to follow a ballistic path: we could have portable mortars (slow fire rate, but large splash damage; can't run while fire or reloading, but very long range; bolts fly a ballistic path, so one can hit monsters on the inside of a crowd) I am all for some more variety if it can be done without tweaking the system too radically.
Bows and arrows will require both the players hands to use, and would be too complex. When firing, there will probably be a longish delay.
I propose you change this original idea to Crossbow. It's more realistic ^^. Single arm fire, and standard reload.