"Is it possible to create a Rule to automatically delete Sent Items older than a specified date (for example all Sent Items older than 3 months)?"
Yes, there are two ways to do this using native Outlook features.
You can use a rule, but will need to run it using "Run Rules Now" and you'll need to update the date each time you run it. Create an 'after message arrives' rule, using the condition 'Received before' and select the date. After adding the desired action, run it on the Sent Items folder and any others where you want to remove older messages. The Received condition supports both before and after so you can use it to process mail that arrives before or after specific dates or between the dates.
If you want automation, use Auto Archive instead. Set the archive options on the Sent Items folder to move or delete mail older than 3 months and configure Auto Archive to run on a schedule. You can use this method to automatically clean out the junk mail, deleted items, and RSS folders too.