In terms of quality, the best way to fix the problems that are bothering you is to re-draw the illustration from scratch using Illustrator or equivalent.
A not-quite-so-good, but still reasonable approach would be to up-rez this image so that one could work on a considerably higher resolution version of it, defining each color area with the pen tool, then re-filling each area with color, always paying attention to sharpness vs anti-aliasing issues at sharp boundaries. Then, as the final step, one could down rez it to whatever size you want and it would remain sharp.
Both of these approaches are much more labor intensive than most good Photoshoppers / pros are likely to do for free.
Unfortunately, there is no magic, one-button-sharpener tool that won't also introduce all sorts of edge artifacts that you'll likely dislike every bit as much as you dislike the artifacts in the version you posted.
However, there are various software tools that one can apply globally (ie, relatively quickly) that may help a bit in certain areas. These include plugins like Topaz de-JPG, Topaz Clean, Topaz Simplify, Power Retouche Pro's Anti-Alias, etc. They won't work as well as using the approaches I mentioned earlier, but they may be good enough for your use and the limited amount of resources (ie, money) you probably want to throw at this problem. Their real advantage of the global tweaks is that these can be applied much much faster than the proper approach. As an experiment, this is what I did.
This approach improved some parts of your image, whereas other areas actually got worse, and in some areas (eg the fade-to-gray on the edges of the white areas) details simply have not been preserved. Depending on how much effort you want to put in, you could blend back in parts of the original in areas where this shotgun approach made matters worse, keeping only the areas that were improved by it. That's your business if you want to do something like that.
HTH,
Tom
PS - Don't forget to click the forum's low rez in-line preview to access the full resolution un-compressed version of my tweaked version.
