Cheapest and easiest is to be satisfied with a 223AI. Simple reamer job, any gunsmith can do it, typically cost about $100-150 for this work.
Rebarreling and converting from 223 to 220S wouldn't need a new bolt, but would need the bolt face opened up. Not difficult or costly, but does carry extra expense over a standard rebarrel. Not sure, but I believe you need a new follower, if not new mag box as well.
Rebarreling shouldn't cost you $1000-1200, that's crazy high. Crapshoot's cost must include a pretty extensive action blue printing as well. Typically, you should be able to get into a barrel for $350 (Crapshoot's using a high dollar carbon wrapped barrel), get it spun on and finish chambered for about $150 with no blueprinting - that should have you all in between $500-600. Again, you'll have additional cost for swapping the mag box and follower, as well as the extra cost for opening the bolt face - but the whole shebang shouldn't cost you anywhere near $1,000 unless you're having them blue print the action along the way (which isn't a bad idea). I'd venture opening the bolt face, replacing the mag box and follower (about $50 in parts), installing and finish chambering a $350 barrel should get you somewhere around $750 with no other blueprinting done.
I've rebarreled Ruger's myself, even buying the tooling I wasn't $1100 into the first one.