Adding simple (one button) Audio player using Custom Fields?

At the most basic, your page will need to have a hidden <audio> element and some javascript to trigger the playback.

You can simply add the URL of the audio file for each post to your custom field, then call get_post_meta() in your template to bind it to the player. Then on the frontend you can bind a click handler to your button that plays the pre-defined audio element.

The custom field (named audioURL in this example)

http://absolute.url/to/audio/file.mp3

In your template file

<audio id="audioPlayer" src="https://wordpress.stackexchange.com/questions/244689/<?php get_post_meta(get_the_ID(),"audioURL', true); ?>" preload="auto">

The JavaScript

// Vanilla JS
document.getElementById('yourButtonID').onclick = function() {
    document.getElementById('audioPlayer').play();
}

// Or if you're using jQuery
$('#yourButtonID').on('click', function() {
    $('#audioPlayer')[0].play();
})

Hope this helps!