One way to do it is to pass the $out handle into the printdata function. Beware of commata after $out in this example:

use strict;
use warnings;

sub printdata {
    my $out = shift;
    my $data = shift;
    $$data{"User-Agent"} //= "--";
    print $out join "|", @$data{ ( "Arrival Time", "From", "To", "User
+-Agent" ) };
    print $out "\n\n";
    %$data = ();
}

open my $out, ">", "tmp.txt";

my %data;
my ($item, $value);
print $out "Arrival Time|From|To|User-Agent\n\n";
while(<DATA>){
    chomp;
    next unless ($item, $value) = /^(.*?): (.*)/;
    printdata $out, \%data if $item eq "Arrival Time" and %data;
    $data{$item} = $value;
}
printdata $out, \%data;
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Alternatively, you could assemble a string in printdata, return it from the sub and print later.