PatchMyPC does indeed work with SQL Server Express. My issue was due to the Schema folder missing from C:\Program Files\Update Services\. Cody sent me the link to this article, which contains a link to download the Schema folder.
Could not find a part of the path 'C:\Program Files\Update Services\Schema\baseapplicabilityrules.xsd' - Patch My PC
Could not find a part of the path 'C:\Program Files\Update Services\Schema\baseapplicabilityrules.xsd' - Patch My PC