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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The while-loop is stepping through all the elements of  @primes and appending the element to a string while that element is less than or equal to the  $n limit. When an element of  @primes is encountered that is greater than the  $n limit, the while-loop terminates.

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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or maybe:

for my $element (@primes) {
    last if $element > $n;
    $answer .= " $element";
    }
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See the last loop control built-in function. Note that the loop exit test in the second variation is explicitly  > whereas this had been only implicit before. Both of these approaches relieve you of the need to manage the  $i index variable.

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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Note that the  $answer variable is no longer needed.

(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.