How To add custom radio boxes to WooCommerce Billing page and change total price by this field?

Try it

add_filter('woocommerce_billing_fields', 'custom_woocommerce_billing_fields');

function custom_woocommerce_billing_fields($fields)
{

    $fields['billing_options'] = array(
        'label' => __('Ordering Method', 'woocommerce'), // Add custom field label
        'placeholder' => _x('Your NIF here....', 'placeholder', 'woocommerce'), // Add custom field placeholder
        'required' => true, // if field is required or not
        'clear' => false, // add clear or not
        'type' => 'radio', // add field type
        'class' => array('form-group'),    // add class name
        'options'         => array(
        'privato_cittadino'         => 'Privato cittadino',
        'azienda_professionista'    => 'Azienda o libero professionista',
      ),

    );