You're halfway there. Get the access time of the file with stat -- that number is seconds since the epoch. You can pass that to localtime and get the date (in list context):

my $atime = (stat($filename))[9];
my ($day, $month, $year) = (localtime($atime))[3, 4, 5];
$year += 1900;    # localtime year is offset by 1900
my @m_names = qw(Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec);
$month = $m_names[$month];
print "$filename last accessed on $day $month $year.\n";
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