You should dereference your array refs:

use warnings;
#use strict;

$colnames = [
          'MEASURE',
          'CNT'
        ];
$arrref = [
          [
            '# PLTADM Queries (ex batch) ',
            '0'
          ]
        ];

my @colnames = @{ $colnames };
print "A size:" . @colnames . "\n";
for ($i=0; $i < @colnames; $i++) {

     print "A: " . $i . " : " . $colnames[$i] ."\n";
}

for ($i=0; $i<= @arrref; $i++) {
      print "B size:" . @{$arrref->[$i]} . "\n";
    for ($j=0; $j< @{$arrref->[$i]}; $j++) {
        print "B: " . $i . " : " . $j . " : " . $arrref->[$i]->[$j] . 
+"\n";
    }    
}

__END__

A size:2
A: 0 : MEASURE
A: 1 : CNT
B size:2
B: 0 : 0 : # PLTADM Queries (ex batch) 
B: 0 : 1 : 0
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Tip #1 from the Basic debugging checklist: strict. See also:

• perldsc