Or if you don't like using map:

#!/usr/bin/perl

use strict;
use warnings;

my @old_array = (
    {
          'is_selected' => 1,
          'name' => 'Me',
          'user_id' => '1'
    },
    {
          'is_selected' => 0,
          'name' => 'Admin Admin',
          'user_id' => '14'
    },
    {
          'is_selected' => 0,
          'name' => 'Proximate Shine',
          'user_id' => '15'
    }
);


#
# sort routine to sort by 'name' key in hash in array
#
sub by_name { ${$a}{'name'} cmp ${$b}{'name'} }

my @new_array;
for my $hash (sort by_name @old_array) {

    push (@new_array, $hash);

}

for my $hash (@new_array) {

    for my $key ( keys %{$hash} ) {

        print "$key => ${$hash}{$key}\n";

    }
    print $/;

}
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