Maybe he was hiding in a cave before that.

This doesn't necessarily imply he has been active for only a month. It may imply that his appearances have increased significantly in the past month. Batman may have been active already for a year or more, perhaps appearing only once every few months or so. As Bruce has grown more experienced in the Batman role, he is likely going out on patrols and missions as Batman more frequently.

Yeah, it's just that the public hadn't seen him much, hence the wild myths. "I hear he drinks blood, I hear he can't be killed" etc.

@DanDare said:

Maybe he was hiding in a cave before that.

The Batcave, of course.