Ok , I think you meant to write

$mtime=(stat ( $ARGV[0] ))[9]; # mtime is 9 , atime is 8!!
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That is to say , stat the file given as the first (index 0) argument to your script, assigning the eighth element in the stat list to the variable $mtime.

have a go with this....

#!/usr/bin/perl -w
use strict;

my $mtime = (stat ( $ARGV[0] ))[9];

print "$mtime\n";

open ( F , '>>' ,$ARGV[0] );
print F 'japh';
close F;

utime $mtime, $mtime, $ARGV[0];

$mtime = (stat($ARGV[0]))[9]; ## Update, this is supposed to be a 9 RE
+ALLY
print "$mtime\n";
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Of course , anything else modifying this file that DOESN'T reset the timestamp is going to throw an irish-screwdriver in the works.
that snippet seems to work w/ perl5.6.1 on rh7.3