How do I determine the ARN of my Amazon route53-hosted zone?

This is addressed in Controlling User Access with IAM, specifically in Route 53 ARNs:

Resource is either hostedzone or change, and ID is the ID of the hosted zone or the change.

The following are examples of a hosted zone ARN and a change ARN, respectively.

arn:aws:route53:::hostedzone/Z148QEXAMPLE8V
arn:aws:route53:::change/C2RDJ5EXAMPLE2

You can use wildcards (*) in place of the ID. [...]

The requested ID is listed in column Hosted Zone ID in the top level Hosted Zones summary of the Route 53 section within the AWS Management Console. Alternatively, you can list your hosted zones via an API call as usual (specifically GET ListHostedZones), and the response contains a respective Id element for each HostedZone element in turn.