The apiserver is declared by a static manifest, when that file changes on disk (within a certain window of seconds), I would expect kubelet to notice that change, kill the apiserver container/Pod, and re-launch it with the new config -- just as it is programmed to do via commands from the apiserver itself. If you have multi-master, AFAIK one must also update the matching apiserver's manifests on the other Nodes, after verifying the first change did what you expected.
Your next question would be where do I find this static manifest
The short version is wherever kubelet is looking for them, so I would start with systemctl cat kubelet.service and look for interesting directories in the unit file, and if that doesn't prove interesting, then for sure launching kubelet with --v=100 will show you everything it is trying to do.