Redirect user to login before viewing custom post

You could set up a child theme, and in the single-listing.php template, add conditions around the current code.

<?php
// If the user is not logged in, redirect
if ( false == is_user_logged_in() ) {
    wp_redirect( wp_login_url() );
    exit;
}
// Else the user is logged in; show the listing
else {
    // Paste the regular template code here
}
?>

Everything else will remain public, while just the individual listings will force the user to be logged in before they can view the content. (This will also prevent search engines from seeing the content, so bear that in mind.)