Yes, it’s possible to listen to podcasts on your Stitcher playlists from oldest to newest episode. Here are the instructions for each platform. Note: This feature is not available on the web app.
On the playlist in question click the settings icon (it looks like a gear wheel). Then click the 'Sort' button. You will see lots of options by which to sort the playlist, including by date published.
Go to "my stations" and select the playlist you would like to reorder. Tap the symbol at the top right that looks like 3 bars of varying lengths You will see a few options by which to sort the playlist, including date published, and length of episode.