Good question!
Ideally, you should probably be doing Joomla and third party extension updates and some of the other checks more than once a year.
Joomla
On a regular basis, run a backup and update to the latest Joomla version.
Ideally subscribe to the Joomla Security News feed so you are kept informed of core Joomla security updates and apply any high priority releases as soon as possible.
You should also check for and apply any security hotfixes for Joomla EOL versions if you are supporting any of these (e.g. Joomla 1.5 and Joomla 2.5).
Third Party Extensions
Minimise the number of third party extensions (including templates) where possible. Remove any third party extensions (including templates) that are no longer being used. Update the remaining third party extensions (including templates) to the latest versions.
Ideally subscribe to the Joomla Vulnerable Extensions List (VEL) so new vulnerabilities can be quickly attended to. Scroll to the bottom of the page for the subscription link.
Consider replacing third party extensions that are no longer being actively developed or supported. For example, if you are still using Xmap, you should probably switch to OSMap or similar.
I find managing third party extensions easier by maintaining a list of the installed third party extensions for each website in a spreadsheet with the currently installed version number.
Performance
Disabling or removing third party plugins that are not being used can help improve performance.
I tend to use GTMetrix to check the website is loading in a reasonable time. GTMetrix offers suggestions (e.g. enable GZip) for improvement and the "Waterfall" page provides a detailed list of how long each element takes to load so it is easy to pinpoint issues (e.g. oversize images loaded by your client).
Creating a free GTMetrix account enables you to run the test from various locations around the world and compare how quickly your website loads in the USA compared to Australia (for example).
Consider moving to a better host if the current hosting environment performs poorly. The performance difference between a poor host and a good host can be massive.
Also consider moving the website closer to the target audience if necessary. You may be able to shave a second or two off page load times if the server and target audience are close together geographically compared to them being on opposite sides of the world. Enabling a content delivery network (CDN) may also solve this issue.
It's possible that archiving articles may improve performance although I suspect any improvement would be tiny compared to disabling or removing unused plugins or choosing a good web hosting company, for example.
Database
The session table on busy Joomla websites can sometimes become corrupted and this breaks the website. On websites that are susceptible to this, I install Akeeba Admin Tools Pro, enable the "System - Admin Tools" plugin and schedule the Session Optimizer to run every hour or so. You can also schedule the clean up of expired sessions, the cache and the temporary directory with this plugin.
You can also repair and optimise Joomla tables and clean the temporary
directory using the free version of Akeeba Admin Tools.
Disk Space
I tend to check the /public_html, /administrator, /administrator/logs, /logs and /tmp folders and delete error logs and other extraneous files. Some of these can become quite large and affect the size of your backups.
Assuming you are running backups regularly using Akeeba Backup or similar and copying backup files off-site, you can probably delete most of the the old backup files. I tend to keep the latest backup file on the server in case a quick restore needs to be done.
Security
In Users -> Manage, check for suspicious Administrator accounts and consider deleting unused accounts over a year old or so. The "System - Admin Tools" plugin in Akeeba Admin Tools Pro can automate the deletion of inactive accounts.
Many hacked websites are not obviously hacked and it is prudent to periodically scan your website for malware rather than waiting for your hosting company or Google to find malware which can result in a suspended hosting account and/or a search engine ranking penalty.
I use and recommend mysites.guru (formerly myjoomla.com) although there are other services such as sucuri.net etc.
File and Folder Permissions
Check that the files and folders (except maybe configuration.php) in System -> System Information -> Folder Permissions are all writeable. Sometimes your hosting company will move your account to another location and upset the path to the tmp and logs folders.
PHP Version
At System -> System Information -> System Information -> PHP Version check the PHP version your website is running on is still suitable. Consider updating to a supported version if your version is not supported any more.
A good web host will allow you to select a supported version of PHP in the hosting control panel or similar.
Copyright Notice
Check the copyright notice year is still appropriate. Some websites may need to be updated manually at the beginning of each year.
