cloud run is simple serverless that run your containers, so cache will be purged, as soon as container removed.
Vercel is also serverless, but they manage cache in their own way https://vercel.com/changelog/introducing-vercel-data-cache so it remain live even when containers dead.

do I have to config a custom cache handler then?

according to the docs: Using a custom cache handler will allow you to ensure consistency across all pods hosting your Next.js application.

theoretically yes, but docs is kinda cumbersome, they didn't explain will it handle all cache types or not.

ok I just don't know what to do to make sure the entire caching just works... by know it seems to have some issues. Changing something in a headless CMS (contentful) doesnt trigger revalidate, then I revalidate manually, which fixes the issue and all of the sudden after a couple minutes the old version is being displayed again.