iTunes: How do I export a podcast as an mp3 to my phone?
Podcasts can be synced to your iPhone when connected to iTunes. These will then be available in the Podcasts app on your phone. 

When you have your iPhone connected to iTunes view your device page in iTunes. Then click on Podcasts and from here you can manage your podcasts on your phone.

I am assuming that you are using iTunes and have an iPhone, you didnt state this in your question (although you have added these tags).