First, I don't think the code you posted work as you expect. The the elements of the two arrays (@Name and @Address must be enclosed between parens. Otherwise, you will only get the first element of the array.

Secondly, I find a bit cumbersome the way your datas are divided in two arrays, one for the names and one for the addresses, unless they are provided to you in this way, I would suggest you put all the user information in an array (or a hash), and put all these arrays into a big one (see perldsc for more infos on data structures.)

Now, about XML::Writer, it tend to agree with Mr. Q&A, I don't like the File::IO voodoo thing you have to do, so I prefer just printing the XML to STDOUT.
As for optimizing you XML::Writer code, you could probably use a loop to go through each element of your structure and write the proper dataElements for each name and address.
The final code should probably look like the following:

#!usr/bin/perl -w
use strict;
use XML::Writer;

my $writer = new XML::Writer( DATA_MODE => 1, DATA_INDENT => 4 );

my @Name    = ("Fred Flintsone", "Barney Rubble");
my @Address = ("111 Rock River Lane", "113 Rock River Lane");

my @flinstones;
for (0 .. $#Name){ push @flinstones, [ $Name[$_], $Address[$_] ] }

# which is the same as :

@flinstones = (
    [ 'Fred Finstone', '111, Rock River Lane' ],
    [ 'Barney Rubble', '113, Rock River Lane' ],
);

$writer->xmlDecl( "UTF-8", 1 );

$writer->startTag( "Application");
foreach my $character (@flinstones){
    $writer->startTag("Customer");
        $writer->dataElement("Name",    $character->[0] );
        $writer->dataElement("Address", $character->[1] );
    $writer->endTag("Customer");
}
$writer->endTag("Application");

$writer->end();

__END__
Results:
<?xml version="1.0" encoding="UTF-8" standalone="yes"?>
<Application>
    <Customer>
        <Name>Fred Finstone</Name>
        <Address>111, Rock River Lane</Address>
    </Customer>
    <Customer>
        <Name>Barney Rubble</Name>
        <Address>113, Rock River Lane</Address>
    </Customer>
</Application>
[download]

Oh, and I'll probably lowercase the name of all the XML elements, but that's just me, maybe you can't do that in your application...

Can you please tell me how to redirect the output to a file named xxx.xml without using the IO output redirection?

You can use $writer->dataElement( 'Name', $Name[0]); to generate <Name>Fred Flintsone</Name> and replace 3 lines in your code. This would shorten it and make it a lot more clear.

I tend to dislike using the IO::File thing and write to STDOUT instead (just don't use the OUTPUT argument and redirect the output of your script) but maybe that's just me.