Here's what your instructor may be hinting at. Let's use the re-formatting of AppleFritter:

sub listprimes {
     my $n      = shift; 
     my $i      = 0;
     my $answer = "";

     while($primes[$i] <= $n) {
          $answer .= " $primes[$i]"; 
          $i++;
     } 
     
     return $answer;
 }
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Another way to step through (or "iterate over") each element of an array is with a foreach loop (which can be shortened to for in all cases). This use of a for-loop is a common Perl idiom. Inside the for- or foreach-loop, you can decide to either append the element to the string or to exit the loop. (This doesn't use push; we'll get to that presently.) Something like:

for my $element (@primes) {
    if ($element <= $n) {
        $answer .= " $element";
        }
    else {
        last;
        }
    }
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for my $element (@primes) {
    last if $element > $n;
    $answer .= " $element";
    }
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How could you use push in all this? (I wouldn't bother; the code now seems simple enough.) In any event, maybe:

my @in_range;

for my $element (@primes) {
    last if $element > $n;
    push @in_range, $element;
    }

return " @in_range";
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(But see NetWallah's reply for a very neat and concise implementation of this function.) (Update: But see my caveat.)

Update: Many small wording changes and additions of afterthoughts, and $string changed to $answer everywhere.

TIMTOWTDI is a very good principal but homework assignments often require use of a particular tool, for breadth of learning purposes. If you're going to do the assignment, for heaven's sake let's get a decent grade on the thing.

Dum Spiro Spero

Push is not necessary to achieve the result, but here is how you could use it:

 sub listprimes {
    my ($n) = @_;
    my @primes_leq_n;
  
    for my $p (@primes){ # for and foreach are synonymous
       last if $p > $n;
       push @primes_leq_n, $p;
    }
    return join " ", @primes_leq_n;
 }
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