We know that Azure Compute is PaaS so the Operating System (Windows Server 2008 R2) has to be patched and upgraded automatically.
I just wanted to know will there be any downtime during the patching or Compute upgradation...?
If you only have a single instance of a particular VM role, then yes - you'll have a short bit of downtime, as you need to be rebooted. Likewise, if the host OS is patched, you'll have a bit of downtime.
If you run two or more instances, then the SLA kicks in, because your instances are separated into different containers/network branches/etc. These are fault domains. So even if a network segment, router, or entire rack were to go offline, you'd have another instance somewhere else.
During OS updates, your instances are divided into upgrade domains, so that they're not all upgraded at once. This leaves your service in an always-available state, as long as you have two or more instances of your roles. For background processes that aren't customer-facing (say, in a worker role that simply reads from queues and processes queue items asynchronously), you can probably get away with a single instance of that role, provided you can handle the work load and that it would be ok to have occasional processing delays.
See this recent TechNet blog post for more details around fault domains and upgrade domains.