Does search engines like google index unfinished .netlify.app sites?

Custom headers are indeed a feature available at all account levels, and that should be one you can set:

You could also use robots.txt (https://www.robotstxt.org/) for a similar purpose.

Note that you’ll want to be careful not to deploy either to your production builds, since presumably you don’t want google to fail to index those! This pattern may be of assistance: