You can do this by being more general in the exclusion - you want digits following non-digits. eg:

#!/usr/bin/env perl 
use strict;
use warnings;
use Test::More tests => 1;

my $str = '{\"id\":01,\"start_time\":\"1477954800000\",\"stop_time\":\
+"1485817200000\",\"url\":http:://www.example.com\}';
my ($extracted) = $str =~ /start_time\D*(\d+)/g;
is ($extracted, '1477954800000', 'Value extracted');
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