Drupal Core Auto Update? Is there a way to update Drupal Core without reuploading files and following the UPDATE.php file?  Something similar to how you can update a Wordpress install with a click of a button?  I wasn't sure if there is a module that will allow this?

--edit--

I've attempted the "drush up" command per @Enxebre's suggestion.

I installed Drush to my shared host and it seems to be running correctly. The site is still not updating though. I checked for updates and its still at 7.22 instead of 7.23.

I also had to run an update on Drush.
Here are my results from the terminal command:

Drush self update:

drush self-update 
X-Powered-By: PHP/5.2.17
Content-type: text/html

A newer version of drush, 7.x-5.x-dev (2013-Aug-06), is available.  Would you like to back up your current drush, version 4.3-dev, to /home1/munortap/drush-backups/drush/20130912205746 and replace it with the newer release? (y/n): y
&lt;br /&gt;
&lt;b&gt;Warning&lt;/b&gt;:  fwrite(): supplied argument is not a valid stream resource in     &lt;b&gt;/home1/munortap/public_html/drush/includes/drush.inc&lt;/b&gt; on line &lt;b&gt;576&lt;/b&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;b&gt;Warning&lt;/b&gt;:  fwrite(): supplied argument is not a valid stream resource in     &lt;b&gt;/home1/munortap/public_html/drush/includes/drush.inc&lt;/b&gt; on line &lt;b&gt;576&lt;/b&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;b&gt;Warning&lt;/b&gt;:  fwrite(): supplied argument is not a valid stream resource in     &lt;b&gt;/home1/munortap/public_html/drush/includes/drush.inc&lt;/b&gt; on line &lt;b&gt;576&lt;/b&gt;&lt;br /&gt;


Drush up:

[~/public_html/misadventureinc]# drush up
X-Powered-By: PHP/5.2.17
Content-type: text/html

&lt;br /&gt;
&lt;b&gt;Notice&lt;/b&gt;:  Use of undefined constant STDERR - assumed 'STDERR' in         &lt;b&gt;/home1/munortap/public_html/drush/includes/drush.inc&lt;/b&gt; on line &lt;b&gt;1668&lt;/b&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;b&gt;Warning&lt;/b&gt;:  fwrite(): supplied argument is not a valid stream resource in     &lt;b&gt;/home1/munortap/public_html/drush/includes/output.inc&lt;/b&gt; on line &lt;b&gt;34&lt;/b&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;b&gt;Notice&lt;/b&gt;:  Use of undefined constant STDERR - assumed 'STDERR' in     &lt;b&gt;/home1/munortap/public_html/drush/includes/drush.inc&lt;/b&gt; on line &lt;b&gt;1668&lt;/b&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;b&gt;Warning&lt;/b&gt;:  fwrite(): supplied argument is not a valid stream resource in     &lt;b&gt;/home1/munortap/public_html/drush/includes/output.inc&lt;/b&gt; on line &lt;b&gt;34&lt;/b&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;b&gt;Notice&lt;/b&gt;:  Use of undefined constant STDERR - assumed 'STDERR' in     &lt;b&gt;/home1/munortap/public_html/drush/includes/drush.inc&lt;/b&gt; on line &lt;b&gt;1668&lt;/b&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;b&gt;Warning&lt;/b&gt;:  fwrite(): supplied argument is not a valid stream resource in     &lt;b&gt;/home1/munortap/public_html/drush/includes/output.inc&lt;/b&gt; on line &lt;b&gt;34&lt;/b&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;b&gt;Notice&lt;/b&gt;:  Use of undefined constant STDERR - assumed 'STDERR' in     &lt;b&gt;/home1/munortap/public_html/drush/includes/drush.inc&lt;/b&gt; on line &lt;b&gt;1668&lt;/b&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;b&gt;Warning&lt;/b&gt;:  fwrite(): supplied argument is not a valid stream resource in     &lt;b&gt;/home1/munortap/public_html/drush/includes/output.inc&lt;/b&gt; on line &lt;b&gt;34&lt;/b&gt;&lt;br /&gt;
If you like you can use the CMS Updater module to enable automatic Drupal core updates. Find the module on drupal.org here: https://www.drupal.org/project/cms_updater

If you need more information about the service behind, lokk here: https://cms-updater.com

Excerpt from project page:


  CMS Updater module helps you to keep your Drupal site secure at all times.
  Right after a new Drupal core release is published, this is recognised by the CMS Updater Service, which in turn will update your Drupal system.
  
  There is no need for intervention on your part – just check your mailbox from time to time, read your PDF update, and see if the automated updates have been successfully completed (or not).
  
  The use of this module requires the purchase of a CMS Updater licence key for € 29.99 incl. VAT. This fee covers one year of updates - your subscription will not be automatically extended, although you will be reminded when your subscription is close to expiring. You can buy the licence key directly in your Drupal admin area under /admin/config/system/cms-updater. If you need a hand or more information, visit https://cms-updater.com.


Emphasis on price added for purpose of this answer.