I fear I cannot contribute a lot in my extremely limited time. Is there a 'getting started' guide somewhere?

Honestly you don't even have to submit a single patch to contribute.

Here is the developer's FAQ: https://wiki.postgresql.org/wiki/Developer_FAQ

I know the feeling of having extremely limited time, i'm in the same boat. But I use Postgres every day with my startup, and we lean very heavy on the database (not by volume of data, just by complexity of the problems). I've reviewed a few patches that I wanted to see get into Postgres, and contributed to discussions on the mailing list. That's about the extent I am able to do at the moment, but it's better than nothing. More reviewers are never a bad thing.