How to set “Site Address (URL)” programmatically on WP multisite?

If you just want to update that one value, then you can use wp_update_site():

wp_update_site( 123, [
    'domain' => 'new-slug.domain.com',
] );

But to really update the site URL/address, you’d need to update both the siteurl and home options for the site:

switch_to_blog( 123 );
update_option( 'siteurl', 'new URL here' );
update_option( 'home',    'new URL here' );
restore_current_blog();

And here’s a full working example which does both the above mentioned steps:

$main_net = get_network( get_main_network_id() );
$site_id  = 123; // ID of the site you're updating

// Update the site domain.
$new_domain = 'enter-slug-here' . '.' . preg_replace( '|^www\.|', '', $main_net->domain );
$site_id2   = wp_update_site( $site_id, [
    'domain' => $new_domain,
] );

// And then the site URLs.
if ( $site_id2 && ! is_wp_error( $site_id2 ) ) {
    $details     = get_site( $site_id );
    $new_scheme  = parse_url( get_network_option( $site_id, 'siteurl' ), PHP_URL_SCHEME );
    $url         = $new_scheme . '://' . $new_domain;

    switch_to_blog( $site_id );
    $url = untrailingslashit( esc_url_raw( $new_scheme . '://' . $details->domain . $details->path ) );
    update_option( 'siteurl', $url );
    update_option( 'home',    $url );
    restore_current_blog();
}

Additional Notes

1. Make sure the new domain isn’t a reserved word.

   Update Apr 04 2020: That’s actually just for sub-directory Multisite installation.

2. Make sure the new domain isn’t an existing username.

   Update Apr 04 2020: Please ignore that, I just didn’t really notice what
   this
   is for. And wp_update_site() would return an error anyway, if the
   domain already exists. 🙂

3. This answer is for subdomain Multisite installation only.