Well, you could declare $x within the call to lambda, instead of surrounding it:

#!perl
use strict;
use warnings;

sub lambda {
  my $fn = pop;
  my @foo = @_; # foo references $x, etc.
  sub {
    # Put values into referenced params
    ${$foo[$_]} = $_[$_] for 0..$#_;
    &$fn;
  }
}

my $x = 3;

my $closure = lambda do {
  my ($x, $y);
  \($x, $y) => sub { print "Got $x and $y\n" }
} ;

$closure->($_, $_+1) for (1..4);

print "And my X is $x\n";
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Though I suspect that's a distinction that won't be seen as advantageous. It's just as easy to write

my $closure = sub {
  my ($x, $y) = @_;
  print "Got $x and $y\n";
};
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which is the Perlish way to do what the OP wants to do.