This is a missable achievement, and a bit of a frustrating one.

Note: It took me quite a while to put these together, and I'm not the best at either the game or laying out the best screenshots, so I'm sorry for how many times Shantae is shown all over.

First off, the full area map, because why not? It's never included in other people's guides (that I've seen) despite still being visible in-game, so here it is::

Now the individual maps for each of the sections, in order. If you take any damage, you'll "die" and restart the section you are in. This doesn't matter, since if you're going for the achievement you have to do this in a single perfect run since your last save. (Including the whole conversation just before the run starts.)

Section 1:
Section 2:
Section 3:
Section 4:
Section 5:
Section 6:
Section 7:
Section 8:
Section 9:
Section 10:
Section 11:
Section 12:
Section 13:
Section 14:
Section 15:
Section 16:

Section 17: SAVE SPOT!
You finally get a Save Spot! If you've gotten this far in a single no-damage run since your last save, you are safe to save before moving on.

Section 18: Cacklebat
Moving on and out to the right there is one final obstacle, a Cacklebat. Jump over it and there are no more obstacles, just run on to deliver Rottytops. (If you fail the jump, just reload the save you just made after Section 16 and you're still good to go!)

Assuming you did it all correctly, you'll get the achievement after dropping off Rottytops to Abner.