from what i have read in the past you can easily do an update without the updater app and without having shell / sudo rights with a few simple steps.
Keeping old installation files left behind seems to be one of the main issues when upgrading. So you should delete all files while keeping the /data and /config folders. Afterwards just upload the new ownCloud 10.0.10 files via e.g. FTP to your web space (make sure to upload hidden files like .htaccess as well) and then point your browser to your ownCloud URL to start the update routine.
Make sure to do some backups of your files and database before like it is recommended for every software you’re updating.
It still might be required (like some one said in another thread) to have shell access to fix up a few things if something goes wrong. In the future it might be also possible (like also said by some one in another thread) that it might be required in the future to have full root/shell access for ownCloud.