I haven’t found an app with a Catholic Bible on it, but Librivox, an organization of *volunteer *readers, has the New Testament and most or all the Deuterocanonicals, Douay Rheims version The NT is about the 9th one down–the link leads to the page on which are the links to the readings in about 1-hour segments.
You can download to your computer and thence to ITunes then put on your iPhone. If you do this, go ahead and get the zip file of all of them at once.
You can also use Spreadsong’s Audio books app, but that only stores part of each book, which If you are not within WiFi, means you can run out of listening material, so you can use the Downloads Manager app or similar to download the whole thing to.
Just the NT is about 25 hours long, so it may keep you going for a while.
DISCLAIMER: Catholic Answers has turned over the archive to Catholic-Questions.org and no longer owns, manages, or moderates the forums. For additional apologetics resources please visit www.catholic.com.