Alright, this was hell to do, but I finally managed it!
The trick is to be very conscious of power ups and upgrades. The coffee cup isn't that much help, so just ignore it. Avoid using the screen clearing bomb at all cost, because it means you won't get any drops. And you need to collect as many coins as possible to buy the upgrades you very much need!
The upgrades you can buy are always, from left to right: Walking speed (boots), weapon speed (gun), weapon power (ammo box). A salesman shows up at the end of every other level, and you can only buy a single upgrade each time.
The first time you see the salesman, upgrade weapon power. The second time, upgrade weapon speed. The third time you probably won't have money for either of those, so save your money! Then get weapon power again on the fourth time, and weapon speed again on the fifth. Avoid buying the walking speed upgrade, as in my experience it doesn't actually help much.
After clearing each level with enough coins and without dying, exit the arcade game and go back to your house and sleep to save. Then continue the game the next day. Exit if you die or if you don't get enough coins.
Finally, I just want to mention that I found the hardest part of the game to be the second world, with the third world being a bit easier again. And especially the final two levels are super easy.
For the bosses, they're not really a problem: Just take it slow and keep safe for the first two. For the third and final you ought to start the map with either the machine gun or sheriff star upgrade, step slightly to the side, then fire straight down to easily kill the boss.
Good luck!