The best way to install it is by installing the VSCode plugin, it will download the binary for you. rustup may arrive in the future and make things better for other plugins.
I don't use Visual Studio Code, unfortunately. I'm currently using rls through my editor's support for LSP implementations.
You can also compile rust-analyzer yourself. It doesn't need any custom stuff from what I saw, and supports Rust stable.
Thank you for letting me know about that. Last time I gave it try it required building the analyzer and then installing the extension manually. The process is much less tedious now.
It's not part of rustup. At least for the VSCode plugin, it'll prompt you to download when it first runs. As long as you've got a working rust installation it'll work pretty seamlessly in my experience.