I have one similar to yours, only the Tall Sturdy. I use a twin size sheet and clip it to the top (I have a drop pan from the girls original cage I leave on top to help store their stuff). The sheet wraps around sides and back and will help protect the walls, however the floor, not so much.
Currently I use a little broom/dust pan and just sweep as it sits on hard wood. In the past when the cage sat on carpet, I used the little mini vac (hand held type) and just did quick pick ups as needed.
On cage cleaning day, I pulls stuff out from under the cage (also store some of their stuff), and clean all the way back to the wall. I do try to keep their food served near the front of the cage and towards the side where it's easier to reach the floor, to help reduce the mess farther back. Sometimes, they're just messier than others, and clean up time varies accordingly, lol.