I think the approach went astray before line 1 and the logic looks to be in reverse order from what I'd have done.

A sensible first step (suggested in what is left hanging at the end of your code) is to convert the current time to a format that matches the directories, before doing anything with them at all (i.e. why read them into an array unfiltered - put all the filtering in one place and process in a grep/readdir loop without needing an array...)

Update: added comments in Filter subroutine as requested by OPer

# convert current time to same format as dirs
my $now;
{ my @now = localtime();
  $now = $now[5]+1900;
  $now[4]++;
  for ( my $i = 4; $i > 1; $i-- ) {
    $now .= '_';
    $now[$i] = "0$now[$i]";
    $now[$i] = substr( $now[$i], -2 );
    $now .= $now[$i];
  }
}
# get only those directories that match requirements:
my $dir = "parent-of-directories";
opendir my $dh, $dir or die $!;
for my $file ( grep Filter( now => $now, dir => $dir ), readdir $dh ) 
+{
   my $path = "$dir/$file";
   # and go right ahead and process the directory "$path" here in the 
+loop
   # ...
   #
}
closedir $dh;

sub Filter { # put all filter conditions here ...
    my %opt = @_;

    ( -d "$opt{ dir }/$_" )   # is a directory
      && !/^\./               # is also not hidden/special
      && /(\d{4}_\d{2}_\d{2}_\d{2})/
                              # contains date/hour
      && ($opt{ now } gt $1 ) # that is in the past
      ; 
}
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