The Cursor server is hosted at https://cursor.blob.core.windows.net/remote-releases/${version}-${commit}/vscode-reh-${os}-${arch}.tar.gz.
For example, for the latest release, the link is https://cursor.blob.core.windows.net/remote-releases/0.11.8-769e57fc532d17f247b121cdf4b6c37f1cccb540/vscode-reh-linux-x64.tar.gz
For every new Cursor update, you would need to update the version and the commit, which may be a little bit tricky.