You are not passing the -ArgumentList parameter to New-Object, so when attempting to instantiate the specified type it will look for a constructor that takes no parameters. The parameterless constructor of the AzureStorageContext class is protected, not public, though...
protected AzureStorageContext ();
...so New-Object will not be able to invoke it.
That same Microsoft.WindowsAzure.Storage.dll assembly is used by the Azure.Storage package. Upon installing that...
Install-Module -Name Azure.Storage
...you can invoke the New-AzureStorageContext cmdlet to create AzureStorageContext instances...
$AzStorObject = New-AzureStorageContext # Additional parameters needed
Otherwise, there is a public constructor of the AzureStorageContext class...
public AzureStorageContext (Microsoft.WindowsAzure.Storage.CloudStorageAccount account);
...that you could use if you pass a CloudStorageAccount instance...
$AzStorObject = New-Object -TypeName Microsoft.WindowsAzure.Commands.Storage.AzureStorageContext -ArgumentList $myCloudStorageAccount