Currently I have an Amazon ALB set to forward traffic depending on the path:
I've set the rules so that www.example.com points to FrontendTargetGroup and api.example.com to point to BackendTargetGroup.
However, for other reasons I now need to use a network load balancer.
How can I achieve the same path based routing approach with this?
You can't do Layer 7 routing (such as hostname or path-based) with a Network Load Balancer. NLB operates at Layer 3/4.
If your service is HTTP, there are very few cases where an ALB won't work, other than the case where your backend is handling all the TLS rather than letting the balancer offload it. You didn't mention that, so you might alternately want to ask a question to verify that an NLB is required and an ALB won't meet your needs.
