How do you build media Press about you or the product prior to launch?

Hey Shabrina. Maybe that helps:

https://blog.ycombinator.com/getting-press-for-your-startup/
https://www.upbeatpr.com/
https://500.co/pr-for-startups-the-art-of-the-press-release-in-7-steps/

Have never tried it myself (yet), but it looks like there are solid SaaS solutions which help you with lean PR services.