How to add post_author column to custom post type

The $supports argument should include author for the Author settings to be available for the custom post type:

'supports' => array(
    'title',
    'custom-fields',
    'author'
)

‘supports’ (array) Core feature(s) the post type supports. Serves as an alias for calling add_post_type_support() directly. Core
features include ‘title’, ‘editor’, ‘comments’, ‘revisions’,
‘trackbacks’, ‘author’, ‘excerpt’, ‘page-attributes’, ‘thumbnail’,
‘custom-fields’, and ‘post-formats’. Additionally, the ‘revisions’
feature dictates whether the post type will store revisions, and the
‘comments’ feature dictates whether the comments count will show on
the edit screen. Defaults is an array containing ‘title’ and ‘editor’.

Docs for register_post_type()

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