First, be sure to have your production credentials for AWS available on your environment, either using awsenv or exporting then by yourself.
export AWS_ACCESS_KEY_ID=<your access key>
export AWS_SECRET_ACCESS_KEY=<your secret key>
Then, build a fresh version of the website with middleman build
(so you won't upload an outdated version of the website) and upload
the build output with middleman sync.
# Build the static site into `./build`.
middleman build
# Upload everything in `./build` to S3.
middleman sync