Is there a way to add additional CSS styles to the styles css file dynamically?

You can enqueue a different stylesheet only for index.php using conditional tags

The following code should be added in functions.php

/**
 * Proper way to enqueue scripts and styles
 */
function theme_name_scripts() 
{
 if ( is_home() ) 
 {      
      wp_enqueue_style( 'style-name',  get_template_directory_uri() .   "/path_to_file/yourStylesheet.css' ); 
 }
}

add_action( 'wp_enqueue_scripts', 'theme_name_scripts' );