How do I add a custom button next to the "Add Content" button?
You have to provide mymodule.links.action.yml in your module.

This is an example from core for the Add content action button from your question:

node.links.action.yml

node.add_page:
  route_name: node.add_page
  title: 'Add content'
  appears_on:
    - system.admin_content


Reference: Providing module-defined actions