Can I set custom ports for a Kubernetes ingress to listen on besides 80 / 443?

No. From the kubernetes documentation:

An Ingress does not expose arbitrary ports or protocols. Exposing services other than HTTP and HTTPS to the internet typically uses a service of type Service.Type=NodePort or Service.Type=LoadBalancer.

It may be possible to customize a LoadBalancer on a cloud provider like AWS to listen on other ports.

I assume you are using NGINX Ingress Controller. In this case, during installation, instead of doing a kubectl apply in the official yaml like this is one, you can try downloading the yaml and changing the port. The file above, which is used for an L4 AWS ELB, would become like this:

kind: Service
apiVersion: v1
metadata:
  name: ingress-nginx
  namespace: ingress-nginx
  labels:
    app.kubernetes.io/name: ingress-nginx
    app.kubernetes.io/part-of: ingress-nginx
  annotations:
    service.beta.kubernetes.io/aws-load-balancer-proxy-protocol: "*"
    service.beta.kubernetes.io/aws-load-balancer-connection-idle-timeout: "60"
spec:
  type: LoadBalancer
  selector:
    app.kubernetes.io/name: ingress-nginx
    app.kubernetes.io/part-of: ingress-nginx
  ports:
    - port: {custom port 1}
      targetPort: http
    - port: {custom port 2}
      targetPort: https

An alternative is to use a more powerful ingress controller. Here is a list of different controllers. My personal choice is Ambassador. If you follow the getting-started page, you just need to change the service definition for the port of your choice:

---
apiVersion: v1
kind: Service
metadata:
  name: ambassador
spec:
  type: LoadBalancer
  externalTrafficPolicy: Local
  ports:
   - port: {custom port}
     targetPort: 8080
  selector:
    service: ambassador