Thank you for reaching out.
Based on your question above
however question is can we access this management.azure.com via site-to-site link instead of on public network?
I do not think Azure REST API management.azure.com can be accessed via the private network. It will help if you could file feedback for this request on our feedback portal
If it helps in your scenario, you can take a look at this document about routing preference and how your traffic routes between Azure and the Internet.
If you can deploy a VM in Azure with public IP address with Routing Preference set to route via Microsoft global network and then access the VM via Site-Site VPN to perform the Azure REST API call. This way the traffic will stay within the Microsoft Wide Area Network and not traverse via internet.
Hope this helps! Please let me know if you have any questions. Thank you!