I have been using Cloudinary to store images, and I want to change the public ID scheme I've been using. Is there any way to update the current public IDs to new ones using the admin API?

For example, if I have an image with a public ID of 1, can I change it to 2 without re-uploading the photo?

You can accomplish this by using the rename API.
For more information (in Rails):
http://cloudinary.com/documentation/rails_image_upload#rename_images