One way that appears to allow the upgrade to be done without the menu option is to browse directly to: URL http://example.com/administrator/index.php?option=com_joomlaupdate/administrator/index.php?option=com_joomlaupdate substituting of course within your site. If I understand correctly, this allows minor version updates, and for version 2.5.14 will allow updating to version 2.5.28 (at the examplecurrent point in time).com part of
I now also see that a major version update is available to me at the URL for your site URL /administrator/index.php?option=com_cmsupdate
This is on my Components menu as option "Akeeba CMS Update" and on my Control Panel as button "Joomla 3.5.1 is available".
Possibly these options require non-standard extension(s) though, as I appear to have a Component called "Cmsupdate" by author Nicholas K. Dionysopoulos and a Plugin called "Quick Icon - Akeeba CMS Update notification for Joomla! updates" by author "Nicholas K. Dionysopoulos / AkeebaBackup.com".