We have bitbucket server hosted within cooperate network that allows only a push and no pull from any server outside network like Bamboo set up in public cloud.
If I am right the current integration only allows bitbucket server to notify bamboo on code commit and bamboo pulls the code.
Is it possible for bitbucket server to push the updated code to bamboo.
Bitbucket Server has no built-in push support. That being said, you could leverage the popular Repository Mirror Plugin for Bitbucket app to automatically push the code to an externally hosted mirror so that the externally hosted Bamboo instance can pull the code from there in turn.
Obvious mirror targets would be Bitbucket Cloud, GitHub, GitLab or AWS CodeCommit (esp. in case your Bamboo happens to be hosted in AWS, though that's not a requirement of course).
By chance I'm currently exploring a resp. CodeCommit mirroring process for our own use cases and apps, so if you'd like to go down that route, just let me know and I'll try to prioritize a resp. How To.
You can not push the updated code from bitbucket to Bamboo directly. What you can do is to create a Webhook in your repository to notify Bamboo that the code as changed.
If you have any further question, please feel free to create a support ticket at https://support.atlassian.com/contact/