sub settime($){ print "nMetrics>You have entered the settime command\
+n";
                        print "nMetrics>The argument given is $_[0]\n"
+;}
        sub showtime{ print "nMetrics>You have entered the show time c
+ommand\n";}
        sub error{ print "nMetrics>Invalid command line options\n";}

        $myhash{"set time"}=\&settime;
        $myhash{"show time"}=\&showtime;
        $myhash{default}=\&error;


        print "Welcome to nMetrics application monitor\n";
        print "Please type in a command\n";

        label:print "nMetrics>";
        while(<STDIN>)
        {
                chomp;
                if (exists $myhash{$_})
                           {$myhash{$_}->();}

                else
                           {$myhash{default}->();}
        goto label;
        }
[download]

Since you have 'commands' that are more than one 'word', you are going have trouble deciding how to split your user commands from your user arguments.

One idea would be to build a regex from the keys of your command hash, which splits the command from the remainder, then pass that remainder to the dispatch table:

# in init phase:
my $commands = join "|", keys %myhash;
my $command_regex = qr/^($commands)\s*(.*)$/i;

# in input loop:
while(<STDIN>)
{
   if ( m/$command_regex/o )
   {
      my $command = lc $1;
      $myhash{ $command }->( $2 );
   } else {
      $myhash{"error"}->( $_ );
   }
}
[download]

But I think that a much better idea is to make 'set' and 'show' top-level commands with a list of variables that they can set and show. Then a simple m/^(\w+)\s+(.*)$/i can split your commands from arguments, then test $1 for definedness as before, and feed in $2 as args.

Some nits: If "default" is an error, then I would call it "error". Also, %myhash is a terrible name for a variable.

Along with the fine comments by fishbot_v2, I'd also like to point out that your subroutines need not be named. This is the perfect situation to use anonymous subroutines. Here's how I would define your hash:

my %commands = (
    "set time" => sub {
        print "nMetrics>You have entered the settime command\n";
        print "nMetrics>The argument given is $_[0]\n";
    },
    "show time" => sub {
        print "nMetrics>You have entered the show time command\n";
    },
    default => sub {
        print "nMetrics>Invalid command line options\n";
    }
);
[download]