Is there a way to auto approve bamboo remote agent without manually approving it on the bamboo server?
I don't think there's an option to automatically auto-approve agents. Agent approval requirement is an security feature so auto approving any remote agent wouldn't be a security feature anyhow.
That being said, there's an option to disable agent authentication which will effectively mean that any new agent is approved right away -> actually what you're asking for.
You can disable agent authentication by visiting Bamboo administration pages