"a stick, a club," Old English *batt "cudgel," perhaps from Celtic (cf. Irish and Gaelic bat, bata "staff, cudgel"), influenced by Old French batte, from Late Latin battre "beat;" all from PIE root *bhat- "to strike." Also "a lump, piece" (mid-14c.), as in brickbat. As a kind of paddle used to play cricket, it is attested from 1706.
Phrase right off the bat is 1888, also hot from the bat (1888), probably a baseball metaphor, but cricket is possible as a source; there is an early citation from Australia (in an article about slang): "Well, it is a vice you'd better get rid of then. Refined conversation is a mark of culture. Let me hear that kid use slang again, and I'll give it to him right off the bat. I'll wipe up the floor with him. I'll ---" ["The Australian Journal," November 1888].