I would not use cardboard for that purpose. Unless the rabbit has exceptional litter habits from day 1, it will get dirty and wet fast. You would need to replace it every time you clean.
Newspaper would work better to be able to gather up the stuff on the bottom easily. You would still need to clean to bottom of the cage as urine will soak through.
The best thing would be to litter train the rabbit. Then you only need to clean the litter box (and spot clean the cage as needed) instead of the whole cage every few days.
I have a dust pan that I sweep up poops and hay that don't make it into the litter box. It could be used to clean up other stuff too. A dog food scoop is also good for scooping up shavings and such.
Cardboard can be good as a toy, but it won't be good for cleaning.