One possible solution would be to update a flag in your DB and have your script check against the value of that flag.

So you want to run some db script on the first available pod after the deployment.

Will it work for you if you follow the following steps during the deployment creation:

$ kubectl apply -f app-deployment.yml

// This will give some time for the pod to start.
$ WAIT_BEFORE_DATABASE_SETUP="${WAIT_BEFORE_DATABASE_SETUP-120}"
$ sleep $WAIT_BEFORE_DATABASE_SETUP 

// Pick any one pod which is in Running State
$ APP_POD_NAME=$(kubectl get pods --field-selector=status.phase=Running -o=custom-columns=NAME:.metadata.name | grep <deployment-name> | head -1)

$ kubectl exec -it $APP_POD_NAME -- bash -c "/scripts/run_db_updates.sh"