Not automatically. But if you want, you can force a post to appear.
1. Go to the old post on your web browser, and click on the datestamp within the post (the part that says the date). This will cause the post to appear on its own, or as part of a thread.
2. Copy and paste the web address of this old post and paste it into the search box within Mastodon. When you search for it, this will force your server to look for it and display it on your server.