I'm in the process of putting a custom arcade stick together for my 360. I've created an R-type image for use as the artwork but all the source material is at 96dpi - I've read that printing at anything other than 300dpi is not going to look too good.
Anything at 200 dpi will be fine... 300 for really nice results, but in all honesty at that point you're grading colors, worrying more about paper stock and humidty....
Exacting standards aside I'd suggest scaling in an app first then printing to just paper to test your results - you may still find them acceptable, if not perfect. If scaling in an app such as photoshop you may try adjusting which scaling algorithm you use for each type of asset.
But really there's no real magic to help; unless you might think the art will vectorise well.
[Just in case you're a bit new to this kind of thing; dpi basically just means pixels per inch]
So the best thing to do is get as high res source material as possible.
I did a test print with the image I created and it actually came out looking ok. I then vectorised it just to see what difference it made and actually liked the results more than the raster image.
99% of the source material available online is 96DPI. I'm no artist unfortunately so creating my own from scratch wasn't an option.
I'm using photoshop and the results from scaling were acceptable - the artwork being based on arcade graphics probably helps matters though!
The 'understood' dpi of a 'screen' is often referred to is 72dpi. But as you may have guessed this is simply bollocks since it depends upon the size of the screen and the number of pixels in its native res. And most sensible on screen applications ignore this - ie. 100% size = 1 image pixel to 1 monitor pixel. Not 100% size = x amount of cm on your screen.
[96 dpi is the new 72 dpi - analists still argue over whats 'standard', it probably won't be of any interest to many people until they consider 150 dpi as the standard screen dpi ... it really only matters for media types trying to automatically load and scale images to dynamic sizes on particularly small/large screens]
In terms of printing if an image is 12inches long @ 100 dpi it can be perfectly tranlated 1:1 to 6 inchs long @200 dpi. They both use is the exact same 1200 pixel long image. Its just formatted in properties to either be 12 inches long or 6 inches long for printing.
So as long as the source images have enough pixels - it really doesn't matter what dpi it says it is - you can easily manipulate it to what you want.