How to erase pattern outside of outline

… or use Draw inside mode (shown in your other thread --> How do I insert pattern into multiple outlines at once )

do you need to remove what's overhanging? is hiding it an option? if so place the circle on top of the rectangle, select both pieces and go to Object->Clipping Mask->Make or press Ctrl+7

If that is a pixel artwork, then the eraser will not work. What you can do is put the circle on top of that Paisley pattern, select both and Object > Clipping mask > Make