Connecting api management with ingress controller with private ip
It's not always mandatory to have public IP for Ingress Controller.
Instead, you can integrate your APIM instance into same kind of vnet as the cluster and configure it for private communication.
As per the Microsoftdoc using the Standard v2 or Premium tier of APIM, which supports virtual network (VNet) integration necessary for private communication is possible.
Refer:
https://learn.microsoft.com/en-us/azure/aks/concepts-network-ingress?source=recommendations
How to connect api management with ingress controller with private ip? - Microsoft Q&A