I've not seen an answer for this mass 301 redirect issue specifically
Well, no, you're not going to find a thread that says how to redirect the exact URL
123-rest-of-name
to the exact URL
rest-of-name
But there are hundreds of posts detailing RewriteRules that will do the trick.
If you already use mod_rewrite (RewriteRule by that name, regardless of what it does) then any redirects in the same htaccess also need to use mod_rewrite.
If not, you can use mod_alias in the RedirectMatch form. You need RedirectMatch (or RewriteRule) rather than simply Redirect because you'll be setting up a simple Regular Expression and capturing part of the request.
Based on your examples, it looks as if both types of redirect can be collapsed into a single rule.
But now... redirect: /some-category/ID-some-product-name.html (e.g. /mens-sweat-shirts/1543-ichiban-fairly-offensive-sweat-in-grey.html)
to: /some-product-name (e.g. /ichiban-fairly-offensive-sweat-in-grey)
Was that cut short for faster typing, or are you really planning to go extensionless? If so, you will need a second rule or ruleset to go from the new URL to where the content really lives.
Come to that: Are you really changing-- or have you already changed-- the names of the physical files and directories? Or are you just changing the URLs?
A change in filename doesn't mean you have to change the URL. But if the end result is shorter and prettier, it's probably the right way to go.