Is private cloud offering "AWS Outposts" part of billing report?

Yes you are correct, aws outpost billing will also be available under the billing report. You can purchase Outposts servers capacity for a three-year term and choose between three payment options: All Upfront, Partial Upfront, and No Upfront. If you choose the Partial Upfront or No Upfront payment option, the monthly charges will be due over the three-year term. Any upfront and monthly charge will apply when your Outposts capacity becomes available for launching instances. For detailed information regarding the billing i will suggest you to reach out to the AWS Account and billing support team. Refer: https://aws.amazon.com/outposts/servers/pricing/

To contact AWS Support

1. Sign in and navigate to the AWS Support Center. If prompted, enter the email address and password for your account.
2. Choose Create case.
3. On the Create case page, choose Account and billing support and fill in the required fields on the form.
4. After you complete the form, under Contact options, choose either Web for an email response, or Phone to request a telephone call from an AWS Support representative. Instant messaging support is not available for billing inquiries.