You might want to check with your hosting provider. I know with Linode I can "clone" my running server, do the upgrades on it, and then swap the IP addresses of the clone and the original (so now the clone is my main server). It's not the most convenient process, but it should work with minimal downtime.
You could do the same thing without the IP swap trick using DNS, it just take a bit longer to propagate. The real catch is what happens to data that was added or changed between when you clone the server to when you make the switch? Ideally you'd want to put your site in a readonly mode if possible.
You could do the same thing without the IP swap trick using DNS, it just take a bit longer to propagate. The real catch is what happens to data that was added or changed between when you clone the server to when you make the switch? Ideally you'd want to put your site in a readonly mode if possible.