Splunk in Azure Marketplace deploys Splunk instance(s) within a newly created virtual network (vnet). By default, Splunk management port 8089 is open only to the private virtual network for security reasons. You could manually change that to allow traffic from a particular CIDR range or the public internet. From the Azure portal, browse to the network security group which was created by the template for the server(s) in question, and update the inbound security rule named "Allow-Mgmt-From-VNET". Alternatively, you could of course run the C# client from within the vnet.
Hope this helps.