The step from version 8.7 (which is 3 years old) to 8.8.10.7 is a big one. But should be possible. The upgrade function knows all the necessary measures. But you a lot has changed since 8.7 and it is necessary for you to do your part after the upgrade. That means that you need to adjust some settings manually.
Since 8.8.8 the settings of the feed /?feed=podcast are separated from the settings for the main feeds of the blog (e.g. /?feed=rss2).
You can find the settings on the Feed/iTunes Settings page of podPress in the section podPress Feeds.
This section contains also settings for further additional feeds. podPress has added since the early version more than the podcast feed to a blog and since 8.8.8 you have control over these additional feeds. You can activate or deactivate or you can customize them separately. This is e.g. useful if you want to have more than one podcast channel.
But in your case you need to go to this section and control only the settings for the first in this section. These settings are the settings for the feed /?feed=podcast. The settings may be the same as for the main/primary feeds of the blog (but is no longer necessary). In every case the feed should be active (there is a check box at the beginning of this section) and you should not alter the slug name. It should remain "podcast" (otherwise the feed URL would be different).
After you have controlled these settings save the Permalinks (simply save them. It is not necessary to modify them.).
You may stay with 8.7. But I do not recommend this. Some parts of podPress 8.7 are not compatible with WP 2.9.x or WP 3.x.
The better choice is to upgrade podPress. If you have questions along the way, you may ask them here.
Regards,
Tim