At that price range you're going to be extremely limited. If you have the skills to sharpen freehand or the patience to learn you may be able to find a single, decent, medium grit bench stone that would do an okay job, but in terms of bench stones, a $20 stone isn't going to be anything special, and if you are trying to go from dull to sharp, and not just sharp to really sharp you're going to want multiple of them in varying grits to get a satisfactory edge.
A MUCH cheaper solution to this would be to just pick up some wet/dry sand paper from any hardware store and use a flat piece of wood as a backing. With sandpaper you're going to want to make edge trailing strokes as you sharpen, but with the right progression of grits and a little practice you can achieve some pretty stellar results for the money you put into it.
Another cheap option would be a system like the Lansky Crock Sticks. It is a V shaped ceramic rod setup where you just draw the knife down the sticks in alternating strokes, it's about as straight forward as it comes, but the ceramic rods that are included are not very aggressive, so any serious reprofiling or edge repair is going to be very time consuming and frustrating. All in all its a decent system that can produce sone very good edges, but definitely isn't perfect.
Truth be told, I recommend you save up some more money and get a spyderco sharpmaker ($50) and a double sided coarse/fine dmt diasharp bench stone ($40) for reprofiling. For under $100 you can be pretty well set for any sharpening needs you happen to have. And yes, I realize $100 isn't exactly a small amount of money, but believe me, it only gets more expensive from here.
Just please DO NOT buy one of those pull through sharpeners. They will absolutely ruin your edge.