So, you're saying this would work then? http://www.linuxmint.com/edition.php?id=185. It says it has 586 by default.Best way to find out would be to try it, but I don't see any concrete reason why it wouldn't work.
Though if the default version of Linux Mint doesn't work you could try the LMDE2 32bit variant which supposedly has compatibility with 486 but with a few limitations such as it not recognising more than one CPU core which may or may not be what you want depending on what CPU you have.