use Encode;  # add this near the top

my $sqlstatement = 'select name_arabic from data';
my $sth = $dbh->prepare($sqlstatement);
$sth->execute || die "Could not execute MySQL statement: $sqlstatement";

my @row;
my @cases; # if handling only one field, let's make this a plain array

while (@row=$sth->fetchrow_array) { # explicitly convert to utf8 (die on failure)
    push @cases, decode( 'cp1256', $row[0], Encode::FB_CROAK );
}
$sth->finish();

# optimize a little: the insert statement only needs to be prepared once

$sqlstatement = 'insert into newdb.names (arabicname) values (?)';
$sth = $dbh->prepare($sqlstatement);

for my $i (0..$#cases) {  # now explicitly convert back to cp1256
    $sth->execute( encode( 'cp1256', $cases[$i] ))
         or die "putsql could not execute MySQL statement: $sqlstatement";
}
$sth->finish();