How to install drush on a shared hosting without SSH and without composer?
Rather than using drush, consider writing a custom script you can call instead. 

In Drupal 7 you can bootstrap Drupal from the command line like this:

&lt;?php
chdir('/var/www/YOUR SITE ROOT');

// no time limit!
set_time_limit(0);

// set some server variables so Drupal doesn't freak out
$_SERVER['SCRIPT_NAME'] = '/importer.php';
$_SERVER['SCRIPT_FILENAME'] = '/importer.php';
$_SERVER['HTTP_HOST'] = 'somewhere.com'; 
$_SERVER['REMOTE_ADDR'] = '127.0.0.1';
$_SERVER['REQUEST_METHOD'] = 'POST';

// change to the Drupal directory
define('DRUPAL_ROOT', getcwd());

 // Drupal bootstrap throws some errors when run via command line
 //  so we tone down error reporting temporarily
error_reporting(E_ERROR | E_PARSE);

// run the initial Drupal bootstrap process
require_once('includes/bootstrap.inc');
drupal_bootstrap(DRUPAL_BOOTSTRAP_FULL);

// act as the first user
global $user;
$user = user_load(1);
// add your custom code here


You will need to tweak this to your specific needs. Place it outside your site root for security reasons.