That's actually not a very large patch... 20 square feet is on the order of 5 feet wide by 4 feet long... unless you mean 20 feet square, in which case you're looking at 400 square feet... truly a daunting job.
Assuming your first estimate is correct, it would be a short bit of work to spade it up, turn it over level and smooth the clods, removing any grass or weed clumps, re-seeding with a mixture of shade and sun grass seed (don't skimp on the seed), rake it in and water daily, never ever letting it dry out. For really good results see if you can locate (probably a farm supply type store) about a pound of oats seed... the kind a farmer would sow and plant. Plant it with the grass seed (more sparingly, probably just a couple of handfulls). It makes a really good 'nurse' crop to shade the new grass seedlings and draw moisture to them. After the frist mowing... voila! it's gone! Best of luck!