API Management does not integrate with AKS directly, so in this scenario you would need to have either an ingress controller, or to expose your service on a node port on a load balancer or similar. You can do this on a private IP so it is not exposed directly to the internet.
For API Management to then access this service you would to use the Premium SKU and join it to the vNet, this would then allow APIM to access the private service in AKS and expose it to the outside world.
An alternative would be to use Azure Application Gateway, if you don't need all the features of APIM. App Gateway can act as a Kubernetes Ingress Controller, see here and is also joined to the vNet.