I've copied a single site from a live multisite setup to my local (MAMP) server. The content is showing from the site however it is unstyled (and also the home page is showing the 404 page). I assume there's some changes I need to make to the DB to fix this and allow the site to function correctly.
I'm running version 6.22
Since your content is showing, your DB connection is OK.
Even if you don't have DB access to the multisite, you should have DB access to your local (MAMP) server. And even if that site's styling is broken, you should still be able to back up its contents using Backup and Migrate.
The fact that the content is unstyled and 404 shows on the home page indicates that your basic site setup is not correct. IIRC, Drupal 6 did indeed appear unstyled if you migrated content between sites without resetting to the default (Garland) before migrating.
There may be a more direct way to repair this (I think Mołot's link in his first comment points to such a recipe), but the way I got around this problem was by setting up an empty single Drupal 6 site with the default D6 theme (Garland). While it was still empty, I configured it and installed and enabled all the modules I was going to need. Then I used Backup and Migrate to backup my content, and migrated that content (using Backup and Migrate) to the newly created and empty Drupal 6 monosite.
After the content was in place, I installed the the theme I wanted to use, and made it the default.
(I.e. the idea is to use Garland until everything else is OK.)
Change you DB details in your settings.php file in sites/all/default directory
