When I work one of our projects. I have experienced the following error:
Directory does not exist.
Parameter name: directoryVirtualPath
NOTE: Get the latest Sitecore databases from your CI or Release environment which already installed the required Sitecore modules.
The error message describes there is a required directory (folder) missing in your solution.
If you have copied the Sitecore correctly this can be due to not coping a required Sitecore module.
Go to the config files of your solution to figure out what are the Sitecore modules that used in your solution.
Since the solution is not working and you can't access the Sitecore desktop to install modules, extract required module and copy the files and folders inside \package\files and paste to your website root.
Since your database has the required items of the Sitecore modules, copying the files of the module is just enough.
This should do the work.