An acrobat is a person skilled at high-wire or other high-in-the-air exploits, while "batty" is a slang term for "crazy" (as is "Bugs").
Acrobatty Bunny is a Warner Bros. cartoon released in 1946 as part of the Looney Tunes series, directed by Robert McKimson (his second), and starring Bugs Bunny and Nero the Lion.