There are so many different ways to update your stand alone ESXi server.

So what way do I use?

I don't like using GUI's at the best of times. I also don't want to use an ISO image to update my install. I could use the vSphere CLI interface or I could use SSH.

SSH sounds like a better idea for my needs, I don't need to install anything like the vSphere CLI. It also means I can apply updates and re-boot the box at a later stage. This method only has one down fall, the ESXi box must be able to make HTTP requests outbound directly, a proxy will not work.

Also ensure you have SSh access to your ESXi install enabled, as all the below steps are done on the ESXi host itself.

Enable http access out from the ESXi box :

~ # esxcli network firewall ruleset set -e true -r httpClient

Next step is to list what updates are available to the host. Run the command :

~ # esxcli software sources profile list -d

The result of that command should look like this : iHost patch listing

You can see from the above the latest patchset is "ESXi-5.1.0-20130402001-standard". This will be the patch set you will now install.
You will always want to install the "standard" patch sets, not the "notools" version or the "s" versions.

To install the above patch set run the command :

~ # esxcli software profile install -d -p ESXi-5.1.0-20130402001-s

Once that has compleated you will see the following : ESXi patchs applied

You will notice that a reboot is required to update the install. You can do this reboot when ever you like. Now you can apply patch's when ever you need to, easily.