Well,
Azure as such does not apply any restriction on the outgoing traffic (beside some bandwidth throttling based on the Size of the VM you are running on, but this quite generous). So in practice you can use any
SMTP relay service.
Whether you can use Mimecast specifically, you shall turn to Mimecast support or products page. But there is nothing in Azure that would stop you from using this or that service (any service).
And Yes, you can use System.Net.Mail.SmtpClient
or any other Smtp Client library to talk to external SMTP Service. Whether that external SMTP Service will accept your connection or not, is not subject to any kind of Azure limitation, but sole decision of the service you chose to use.