Head as far left as you can over the crop fields and enter a door into this rest area. Talk to the Mimiga with the fishing rod.
Follow the Mimiga down to the water area and talk to him to be offered an item that he caught and put in his bucket. It's the key for the room just to the left. Head into it and take the teleporter to be thrown in jail.
Talk to the hungry Mimiga in the jail twice to find out that Sue was in here and put a letter in your pocket. Read it from your inventory, and then escape the jail on the right side.
Once making it out of the jail, you can safely take the teleporter you got caught in back the Arthurs house. If you still haven't traded off your Polar Star, use the Booster to get to the very top of the Mimiga village and into First Cave again. Talk to the Hermit at the end and he'll upgrade your Polar Star to the Spur, which is a direct upgrade! The base shot is now as powerful as the level 3 shot, and instead of using the normal level mechanic, hold the fire button down longer to unless a powerful beam!
Then head to this hidden door by going through fake blocks and speak the now-known password to enter.
Talk to Sue's mother and she'll tell you she's building a rocket. She'll borrow your Booster and give you a Mimiga mask so the Mimiga will talk to you. Go talk to this Mimiga to find out his Sprinkle is broken, then grab the broken sprinkler.
Take it to the Mimiga with glasses in the rest area to replace it with a new one. Instead of returning the sprinkler to the farmer, take it to Sue's mother. You'll then be told to go and fetch Itoh, the Mimiga whom you frightened when entering from the Outside wall.
Talk to Itoh to be given a controller, then return to Sue's mother with it. The rocket will take a bit, so head down and take a sleep in the bed.
Once you awake the rocket will be done outside. Talk to Sue's mother to switch out the Mimiga mask for your Booster, then use the Rocket to launch way up to the Doctor's Throne.