I would simply loop over the elements comparing each stringlength with the current minimal length.

While using all the bells and whistles with map and shift and sort and such, I think it's pretty overdone. Just as plain as this untested:

use strict;
use warnings;

my @array = qw( hello, superduper, hi );
# You know that those commas are part of the strings!?

my $shortest= $array[0];
my $shortest_length= length $shortest;
for ($i= $#array; $i; --$i) {
    my $length= length $array[$i];
    if ($length < $shortest_length) {
        $shortest_length= $length;
        $shortest= $array[$i];
    }
}

print $shortest,"\n";
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Of course, one can shorten this, but you wanted a direction, not a "nice" solution...