There may be other ways (for example, using touch directly), but this is the first way that I thought of.

use Time::Local;
my $file = "foo";

my($atime, $mtime) = (stat $file)[8, 9];
my($sec, $min, $hour, $mday, $mon, $year) =
    (localtime $mtime)[0..5];
my $on_hour = timelocal(0, 0, $hour, $mday, $mon, $year);

utime $atime, $on_hour, $file;
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If you also want the last access time set to 13:00 (in your example), change the last line to

utime $on_hour, $on_hour, $file;
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