Currently, there is no alternative available to the service. You may want to try using the Web App services but the limitations would do you no good.
The cloud services are hyping up in the market right now and it is difficult to find a successor to Azure right now.
Azure service fabric is however recommended to shift over to now for the cloud service, you can get a more detailed explanation about migrating over here: https://docs.microsoft.com/en-us/azure/service-fabric/service-fabric-cloud-services-migration-differences
Before Azure fabric came up, Azure virtual machine extensions were being used, but they aren't of much use now unless you want to exclusively use a resource for post-deployment configuration and automation tasks on Azure VMs.
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