We're starting from the top of tree, searching node with cat_parent_id = '', and then we descend deeper and search chidren of that found parent (cat_parent_id = '$k'), returning array of found children from top to bottom, so in the end you have nice array that you join or use it other way.

use strict;
use v5.10;

my $hash = {
  '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'
  }
};

sub rec {
  my ($hash, $parent) = @_;
  my @str;
  foreach my $k (keys %{ $hash } ) {
    if ($hash->{$k}->{cat_parent_id} eq $parent) {
      push @str, $hash->{$k}->{cat_name};
      push @str, rec($hash, $k);
      last;
    }
  }
  return @str;
}

my @arr = rec($hash, '');

say join(',', @arr);
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UPDATE:

Slightly changed recursive function so it won't check categories that was already $seen as parents. We don't know if we could just delete $hash->{$k} after we found it and extract cat_name, $hash could be needed later, so it's only grep { not exists $seen->{$_}}

sub rec {
  my ($hash, $parent, $seen) = @_;
  my @str;
  foreach my $k (grep { not exists $seen->{$_} } keys %{ $hash } ) {
    if ($hash->{$k}->{cat_parent_id} eq $parent) {
      push @str, $hash->{$k}->{cat_name};
      $seen->{$k} = 1;
      push @str, rec($hash, $k, $seen);
      last;
    }
  }
  return @str;
}
[download]