The Persona PR team confirmed the Xbox 360 version of Persona 4 Arena has lag issues. Atlus is looking into a solution, but for now they recommend players select "custom match," "same area," and "good connection" when playing online.
This problem only affects the Xbox 360 version.
Persona 4 Arena still has a week or so until its released in North America. Hopefully, Arc System Works and Atlus will have a patch ready for launch.