You can use free Podcasts apps on iPhone 5. Podcasts app is the easiest way to discover, subscribe to, and play your favorite podcasts on your iPhone.
Learn how to use Podcasts on your iPhone 5. Download the free Podcasts app from the App Store, then browse, subscribe to, and play your favorite podcast.
Learn how to use the Music apps on your iPhone 5. You can use Music apps for playing music and sharing. Continue reading on how to use it.
The iTunes not syncing with your podcasts, its most likely setting an issue in your iTunes, by default iTunes wont sync podcasts out of the box. Follow some simple steps below to fix