Does Codeberg have anything like Gitlab CI, or does it need to be paired with other build tools like Jenkins, TeamCity, etc?
Yes, but you need to manually request it and there are a handful of things that need to be true. See here for them all: https://codeberg.org/Codeberg-e.V./requests#woodpecker-ci
In summary, it needs to be a public repo with FOSS code and the README must match the project’s intended goal.