Just to highlight, ACR is a repository of images, and if you need to rebuild an image, you need to do that outside of ACR.
You may use ADO pipelines
to rebuild the image or you can use ACR Tasks
to rebuild the image. You can trigger an ACR Task to do the build and store the new image in ACR.
As mentioned, you may add a Build&Push step in the pipeline that can automate this in ACR, please see this doc.
Please see, automate container image builds and maintenance with ACR Tasks
See continuous deployment doc, you can enable CI/CD, so that App Service gets notified of a new image and triggers a pull automatically.
Based on my understanding of your scenario, I have shared the options that you could leverage. Please let me know if it helps or/As requested share additional information about your requirement to better assist you.