“Right off the bat” means done in a hurry, without delay. Similar phrases are: lickety-split and as fast as greased lightning.
The phrase likely arose from baseball, where, after a successful hit, the batter takes immediate action and runs to first base.
In newspapers of the 1880s, the expression was used both in relation to baseball, but also in the figurative sense of doing something fast. This example is from the Albion New Era newspaper, 1883: “A person unused to it would net catch one ‘fly’ out of fifty, and as for stopping and holding a hot liner right off the bat, he might as well attempt to gather in a solid shot fired point blank from a Parrot gun.”
Another example comes from the Biddeford Journal, 1888: “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.”
Slang? My goodness! I’m glad that man can’t hear my language when I’m annoyed.
Rainforest Writer 