Two mistakes:

• Use $1 instead of $3 to get the 1st capture (you only have one set of parens).
• Use \s+ instead of the greedy .*

use warnings;
use strict;

while (<DATA>) {
    if (/Name\s+LUN\s+(\d+)/){
        my $lunid = $1;
        print "\n!$lunid!\n";
        print $_;
        exit;
    }
}

__DATA__
Total Hard Errors:          0
Total Soft Errors:          0
Total Queue Length:         0
Name                        LUN 259
Minimum latency reads N/A
[download]

prints:

!259!
Name                        LUN 259
[download]

Aside: you only need to use code tags for your code and data, not for your question: Writeup Formatting Tips