need to aply a different style sheet for a specific page template

How do I embed styling that will affect this template only?

The trick is to hook into template_include. This allows us to see which template is being applied to a page.

Suppose we have a template with a basename of ‘full-width.php’. Inside our template_include callback, we can conditionally enqueue a CSS style only when the template is full width.

function wpse_258048_template_include( $template ) {
  if( 'full-width.php' === basename( $template ) ) {
    wp_enqueue_style( 'full-width', PATH_TO . '/full-width.css' );
  }
  return $template;
}
add_action( 'template_include', 'wpse_258048_template_include', 1000, 1 );