How to access function from outside of a class within this class in WP plugin?

Create a static getter for your class instance:

class DD_Awesome_Plugin
{
    /**
     * Plugin main instance.
     *
     * @type object
     */
    protected static $instance = NULL;

    /**
     * Access plugin instance. You can create further instances by calling
     * the constructor directly.
     *
     * @wp-hook wp_loaded
     * @return  object T5_Spam_Block
     */
    public static function get_instance()
    {
        if ( NULL === self::$instance )
            self::$instance = new self;

        return self::$instance;
    }

    public function add_menu_page()
    {
        add_options_page(
            'DD Awesome PLugin', 
            'DD Awesome PLugin', 
            'administrator', 
            __FILE__, 
            array( $this, 'display_options_page' )
        );
    }
}

And now you get the plugin instance with:

add_action('admin_menu', function() {
    DD_Awesome_Plugin::get_instance()->add_menu_page();
});

Or:

add_action( 
    'admin_menu', 
    array( DD_Awesome_Plugin::get_instance(), 'add_menu_page' ) 
);