Cloud service deployment packages tend to be slow since they are basically a recipe on how to build and configure your deployment. The deployment not only puts the recipe out in Azure (so it can be used again if it needs to move your machine), but also follows the recipe to build out a VM for your Cloud Service (WebRoles/WorkerRoles are platform as a service so you don't have to worry about the OS and infrastructure level like you would if you were using the Virtual Machine Azure product but they do still run in VMs on physical hardware).
What you are looking to do is something that will update the recipe (your cloud service package) and your deployment after it is out and running already ... there is no simple way to do that in Cloud Services.
However, yes you could create a startup script that could pull the site files from blob storage or some other centralized location - this would compare to how applications (fiddler for example) look for updates then know how to update and replace themselves. For that sort of feature you will likely need to run code as an elevated user - one nice thing about startup scripts are they can run as an elevated user - so they can do about anything you need done on a machine (but will require you to restart the instance for them to run). Basically you would need to write some code that will allow your site to update itself. This link may help: https://azure.microsoft.com/en-us/documentation/articles/cloud-services-startup-tasks/
If you have the ability to migrate to WebApps and WebJobs, I would recommend looking into that since that compute product solves your problem really well.
Here is a useful answer of the differences between WebApps and Cloud Services: What is the difference between an Azure Web Site and an Azure Web Role