Drupal Newbie - Upgrade Questions
Yes, the update is manual. (As someone new to Drupal, you may not know about the command line tool drush, which is as close as we get to being a one-click affair. Drush is awesome, and once you get more familiar with Drupal, you should learn it. It makes your work much more efficient.)

Re the previous answer, 'Download these updates' only works for contributed modules and not for core.

The instructions on drupal.org are excellent.  But before you start, here are a couple of pieces of useful information.


Do not get confused by the instructions at the top of the page sending you to upGRADE instructions.  An upGRADE means moving from one version of Drupal to another, e.g., Drupal 6 to Drupal 7, and is a much bigger deal. An upDATE moves you up by minor versions, e.g., Drupal 7.2.1 to 7.28.
Because you are new to Drupal, it would probably be a good idea to make sure the previous developer did not make the classic error of putting contributed modules and themes in with core modules.  When you do the update you will delete all files EXCEPT [drupal root]/sites, which is where contributed modules, and site-specific files live. On your site, go to admin/reports/updates/list. Open another window in whatever file browser you use, and look at the Drupal file structure.  The list of modules in the update list should match up to folders in [drupal root]/sites/all/modules.  If instead you find those folders in [drupal root]/modules, you have bigger problems than can be answered here.  (Same goes for themes.)
If you have a development and/or staging site in addition to the live site, do the updates there first.  If you don't have non-live sites, you should create them.  You can do updates on a live site, but it's much preferable to test them first.


Sorry, long answer.  There's a lot more good stuff on drupal.org, but it can be hard to find.  I usually use Google to search drupal.org.