Is a "listen again" service available for Beats 1?
Yes, as an Apple Music subscriber, I can verify that paid Apple Music subscribers may go back and listen to replays of most Beats 1 shows on-demand. I personally use this feature to catch up on Zane Lowe all the time. It will allow you to go back and listen to shows up to a couple months old. For example, right now I could go back and listen to many shows from December 2015, but not any earlier.

This is part of the regular Apple Music service, so it also gives you access to on-demand tracks from (much of) iTunes, iTunes Radio, and For You.

$5 per month - student

$10 per month - individual

$15 per month - family