First add a child link to each node then traverse by child order:

use warnings;
use strict;

my $data = {
          '986172' => {
                        'cat_parent_id' => '',
                        'cat_name' => 'Category1'
                      },
          '986178' => {
                        'cat_parent_id' => '986177',
                        'cat_name' => 'Category4'
                      },
          '986177' => {
                        'cat_parent_id' => '986176',
                        'cat_name' => 'Category3'
                      },
          '986176' => {
                        'cat_name' => 'Category2',
                        'cat_parent_id' => '986172'
                      }
        };

map{$data->{$data->{$_}{cat_parent_id}}{cat_child_id} = $_}
    grep{$data->{$_}{cat_parent_id}} keys %$data;

my ($node) = grep{!$data->{$_}{cat_parent_id}} keys %$data;
my @chain = $data->{$node}{cat_name};

while ($data->{$node}{cat_child_id}) {
    $node = $data->{$node}{cat_child_id};
    push @chain, $data->{$node}{cat_name};
}

print join ',', @chain;
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Prints:

Category1,Category2,Category3,Category4
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