The term "MP4 Player" is terribly confusing and just a marketing work. The iPod plays MP4 files, it's what the iTunes store sells, but it's not what people would usually call an "MP4 Player".
They're usually a portable video player, so an iPod Touch might be called one, and the name is to try and give the impression that they're the "sequel" to MP3 players.
Even with one of those though where it does work with iPlayer you will still need an internet connection to listen to the shows. So in the house would be fine if you've got wireless, or on the move might work if you get a gadget such as an iPhone that has a mobile data capability.
To listen to stuff on the move and without any internet connection you're going to need to convert the "stream" of data into a file on your computer that you can then copy onto your gadget. And that's the sort of software that NetworkGuy posted about.
You generally don't need anything "special" to work with an iPod if you're wanting to listen to music or radio, iPod's play a wide variety of audio files.
So for example that www.highcriteria.com software would work just fine with an iPod and the standard edition.