Enabling automatic VM guest patching for your Azure VMs helps ease update management by safely and automatically patching virtual machines. More details and supported OS are here : automatic-vm-guest-patching
In this way azure will handle the automatic patching once in a month and need a reboot also as and when needed. While saying that, not the reboot requirement here too. The zero downtime solution can be implemented only with a supported application design which can handle the downtime situation by switching the application access to a different VM, which is planned to patch in a different schedule