As a matter of fact a lot a companies use this kind of DR, and it is commonly called "Cold DR" (http://en.wikipedia.org/wiki/Backup_site#Cold_Sites)
You will only pay for the space reserved for your storage (S3, EBS, Glacier) and Route 53 Zones.
Just start the instances there, migrate the data you need, configure your instances properly and then stop them.
If you are using other DNS service I suggest you to migrate your DNS management to Route53, because then would be able to easily write a script to automatically change the DNS and then start your instances, or use Route 53 Health Check and DNS Failover (http://docs.aws.amazon.com/Route53/latest/DeveloperGuide/dns-failover.html) plus a little bit o script to achieve a more automated solution.
Hope this helps.