You will want to play on a medium ship, as the challenges won't proc on small ones, and a large ship will take longer and be harder. You will also want to have Zen mode enabled, as it will make the whole process easier. A solo game may be preferable, as a dog will prove handy for stunning aliens.
You may also want to have the Watering bot available in the shop for a contingency plan should things go wrong.

Once the game starts, what you want to do is rush through the rooms, cleaning them as fast as you can. Any aliens you encounter will be taken care of by the dog. Don't forget to feed your dog the aliens, as this will get rid of them without the game counting the aliens as being killed. Use tables to store excess aliens. Don't recycle the garbage bags yet, as it will slow you down! By the end of this phase, you should have a clean ship, but likely not any powered rooms aside from the first one or two. P.S: Feel free to use any electronics you happen upon within the ship itself, as the restriction on them only applies to buying them.

Now that you finished phase 1, you should have a clean, if mostly unpowered, ship. Phase 2 is all about maintaining the ship's cleanliness while you gather the funds to buy batteries for each room.
Don't let doors stay closed for long, splash the vents once in awhile, and gather the garbage that is now strewn all over the ship and recycle it. Once you run out of garbage to grind up, switch to recycling pumpkins. If you were lucky enough to find an exercise bike, you can use that instead of the recycler to get money.

This section details how deal with problems that may crop up during an attempt.
Assuming you haven't lost yet, you can still complete the match fine, but if you are willing to forfeit "Less Shopping", you can buy a watering bot to clean you.
That's on you for taking too long, but all of the challenges can still be salvaged, possibly. Splash the lot of them with water and KO any Swimmers with the broom. Hope there aren't any spiky aliens or you will have to forfeit "Less Shopping" and buy a Watering bot.

That's the end of the guide. I hope this helps! If you have any suggestions, let me know in the comments!