semi-OT, but since the Seeker says regexen solved the problem, I suspect a typo in the OP:

/startpattern/ (Ln 9) is NOT the same as /start pattern/ (in #start pattern in Ln 14.

Further, while others have mentioned the explanation for non-interpolation, no-one, OP included, has posted a solution -- so here's one (which omits the -- confusing to me -- script's use of both a file (of unspecified content) and __DATA__:

Data file:

#start pattern
my name is $name
blahhhhhhh
my id is $id
blahhhh
#Endpattern

#newpattern
my name is $name
blash
my id is $id
#endofpattern

#startpattern
my name is $name
blaghhh
my id is $id
#endpattern

#start pattern
my name is fred
blahhhhhhh
my id is 42
blahhhh
#Endpattern

#1064933.txt
[download]

code:

#!/usr/bin/perl
use 5.016;
use warnings;

# 1064933

my $name="guest";
my $id = 1;

$/ = "\n\n";

open(SOURCE, "<1064933.txt") or die "Can't open 1064933.txt, $!";
open(SINK, ">1064933OUT.txt") or die "Can't open 1064933OUT.txt, $!";
while (<SOURCE>) {
    if ( /start pattern/ .. /Endpattern/) {
        s/\$name/$name/;
        s/\$id/$id/;
        print SINK $_;
        next;
    }
}
[download]

output:

D:\_Perl_\PMonks>cat 1064933OUT.txt
#start pattern
my name is guest
blahhhhhhh
my id is 1
blahhhh
#Endpattern

#start pattern
my name is fred
blahhhhhhh
my id is 42
blahhhh
#Endpattern
[download]

Note, however, that while this limits the exploitability of using an eval solution (by restricting the substitutions to the programmer's specs), it's far short of production-worthy. But then, the notion of globally replacing \$name with "guest" and making all \$id evaluate to 1 suggests that the example code has little resemblence to the actual task at hand.