Through the abuse of the package variable @a your routine works after a fashion, however if you Use strict warnings and diagnostics or die it fails. Consider

#!/usr/bin/perl
use strict;
use warnings;
sub push1{
        my $ref=shift;
        my @a=@{$ref};
        my $s=shift;
        my $len=@a;
        $a[$len]=$s;
        return \@a;
}
sub push2{
        my $ref=shift;
        my $s=shift;
        my $len=@{$ref};
        $ref->[$len]=$s;
        return ;
}
my @a=qw( 1 2 3 4 );
push1(\@a,5);
print "Using push1 ", @a,"\n";
@a=qw( 1 2 3 4 );
push2(\@a,5);
print "Using push2 ", @a,"\n";


Using push1 1234
Using push2 12345
[download]

You either need to do something with the return value or act on a reference to the array rather than a copy of it.