ipod classic podcast tab

• Connect the iPod to your computer
• Click on the iPod icon towards the top

and you will then see the Device Settings/Summary pane:

Click on Device/Podcasts (four items below Summary, in the list shown above):

Note the Sync Podcasts option in the right hand panel. Tick Sync Podcasts. In my screenshot, I have chosen Automatically include 3 most recent episodes of selected podcasts. Use the drop-down options to make your selections. Then tap the Apply button for those new changes to be actioned.

Is the Podcasts sub-menu enabled on the device and/or in your library? Does you library include any downloaded episodes that could be synced to your device?

tt2

See Sync your iPhone, iPad, or iPod using iTunes on your computer - Apple Support. If the steps in there don't help you choose podcasts to sync to your device where do they stop working?

tt2

Glad you have it sorted. (Your screenshot was my clue.)