To get rid of index.php, you need to add "ForceVirtualHost=enabled" in [SiteSettings] group in settings/override/site.ini.append.php and modify your .htaccess per following instructions.
Whoops, sorry, didn't notice that you don't have access to virtual host in apache. Not sure if this will work then.
I've had varying results with trying to remove index.php when not having virtual host setup and ultimately I would just get access to virtual host and be done with it.
Change your SiteURL settings in your default siteaccess site.ini.append.php, the SiteURL may not end with '/' neither the siteaccess name, just 'index.php', and then clear the cache. There are already topics about removing index.php from url, I use .htaccess: