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This is straight from A-zon's product image. $15 - I think you can get the concept a bit better from this image. It's simply called a "Woodriver Single Screw Extractor". As you can see at the lower right- this will be the end result. To get started, grab a scrap of wood and bore to about the outer diameter of the above extractor. This way, you have a collar to control the hollow tube saw, which otherwise would wander all over on you since there's no center point to guide it. Then, simply stab in a dowel of the right size, cut flush. Set your bridge with the good screws, use the hole in the bridge to mark/punch dead center, drill a small pilot and voila- like it never happened.
Good luck. If you have any further trepidation, I'll bet you have a friend/neighbor/relative that would gladly help you regain full use of your instrument.