Kubernetes rolling updates will be creating new pods with updates and gradually change old pods by new, and as soon as new pod is ready it will get traffic by round-robin with the old ones.
But what to do in case that I have 2 services one is using another and I need to make updates in them simultaneously? I don't want the situation that old service 1 will make calls to new service 2.
I suppose blue/green deployment strategy suits good as your task solution. Check these guides: multi-service sample or single service sample.
The main idea - your deployment pipeline should wait for successful replicaset creation and then update service selector to match pods with new version.