All you had to do was save the pic to your computer as a .jpeg, open it in MS Paint, and save it again as a .png.
To elaborate a bit on eswube's comments for this not working. I think it's important to explain exactly why this doesn't work and why it will never work.
The issue is that the .jpeg file format uses Lossy compression (versus the lossless compression used for a png). For image formats, Lossy Compression essentially means that to save space instead of the exact color for a given pixel being stored, a jpeg file stores an approximation of what the color is. The practical result of this for pixel art is that when you go to edit the image after the fact any two pixels which were originally the same color may now be entirely distinct colors as there is no guarantee of them having being stored using the same approximation. (This is what causes the artifacting) In addition because only an approximation of the color was stored the information of what the correct color should be for any given pixel has been permanently lost (hence the term lossy compression) and no amount of opening the image and resaving as a .png will ever fix it. The new png will always contain the incorrect approximation of the color data from the jpeg. This is why either the original .png file or an extensive redrawing is required to fix jpeg artifacting.
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