The stat call returns a list with mtime at index 9. Here's an example. I use glob to get the directory contents, but your readdir works, too. I'll pitch in a Schwarzian Transform to reduce the load of stat.

my @sorted =
    map  { $_->[1] }
    sort { $b->[0] <=> $a->[0] }
    map  { [ (stat)[9], $_ ] } glob('*');

{   # diagnostic
    local $, = ' ';
    print @sorted;
}

my @results = grep { /^$mask/ig } @sorted;
print "Your file is: ", $results[$position-1], $/;
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Having $b to the left in the sort comparison gives newest first.

If $mask is given as a glob pattern instead of a regex, that can replace '*' in the glob call to remove the grep step. If the arguments come from the command line, your shell will do the glob for you, making @ARGV[1..$#ARGV] your filename list.

Update: Improved version, quicker by sorting a smaller pool:

my @candidates = grep { /^$mask/ig } glob('*');
my @sorted =
    map  { $_->[1] }
    sort { $b->[0] <=> $a->[0] }
    map  { [ (stat)[9], $_ ] } @candidates;
print "Your file is: ", $sorted[$position-1], $/
  if defined $sorted[$position-1];
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