This is mildly idiomatic (the grep/map, for example, and there is probably an equally terse but less idiomatic version). I hope it's otherwise serviceable and interesting. XML::LibXML and Text::CSV_XS for more fun and deeper options.

Aside: nodeName ne '#text' is more readable but nodeType != 3 is a little more portable (older versions call text nodes "text").

use strict;
use warnings;

use XML::LibXML;
use Text::CSV_XS;

my $doc = XML::LibXML->new->parse_fh(\*DATA);
my $root = $doc->getDocumentElement;
my $csv = Text::CSV_XS->new({ eol => "\n" });

my ( $ip_node ) = $root->findnodes("Header/IpNumber");
my $ip = $ip_node->textContent;

open my $out, ">", "$ip.csv"
    or die "Coulnd't open $ip.csv for writing: $!";

$csv->print( $out, [ $ip, undef ] );

for my $content_element ( $root->findnodes("ContentElement") )
{
    my @elements = map { $_->textContent }
        grep { $_->nodeName ne "#text" }
        $content_element->childNodes;
    $csv->print( $out, \@elements );
}

__DATA__
<someRoot>
  <Header>
    <IpNumber>AC_123</IpNumber>
  </Header>
    <ContentElement>
      <IdNumber>xyxyxyxy-yy</IdNumber>
      <InstanceNumber>001463010000016</InstanceNumber>
    </ContentElement>
    <ContentElement>
      <IdNumber>ceiling-cat</IdNumber>
      <InstanceNumber>77777777777</InstanceNumber>
    </ContentElement>
     <ContentElement>
      <IdNumber>basement-cat</IdNumber>
      <InstanceNumber>666666666666666666</InstanceNumber>
    </ContentElement>
</someRoot>
[download]

If you were using strict (or my code :) you'd see the difference between $InstanceNumber and $IstanceNumber.

But in your code in the handlers start and end you don't empty out the variable $Number, is just because you check if you are in the node with the variable $inHeader = 1; ? B/R