If you have the Instituto Exactamundo DLC (the marketplaces lists it as "perfectly sized katamari"), that's the quickest way to get requests done. You'll need to roll up a little bit before the King will let you hit X to end the stage, but it'll still take much less time than Egg School, which is your best non-DLC option.
Use the start button to skip past the King's dialog quickly.
Time Attack & Eternal rolls DO NOT COUNT. You also CAN NOT replay a stage when it's done, you need to start it all over again from the hub.
You can see how many times you've rolled by going to space and viewing Everyone's Katamari. Problem is, that includes Time Attack & Eternal rolls, so you'll need to do a bit of math there. My advice would be to note the number you have listed here before you begin grinding for this achievement, just so you can follow your progress.