Try this:
<?php
$cat = get_the_field('artist_news');
$recentposts=get_posts('showposts=3&category=$cat');
?>
Not exactly sure how WordPress functions handle variables inside of strings like that so you might need to do this:
<?php
$cat = get_the_field('artist_news');
$recentposts=get_posts('showposts=3&category=' . $cat);
?>
