How to update your vCenter Server Appliance, going from 5.1, to 5.1.0a or 5.1.0b, is easy

Moving from vCenter 5.10.5100 Build 799730 to 5.1.0.5200 Build 880472 was quite straight forward. A few clicks, walk away, come back, it's done. ESXi and all your VMs keep cruising right along, non-disruptive. Mine took about 20 minutes total, and you need only click a few times to kick it off, then walk away. Naturally, since you're updating vCenter itself, you can't have any running vMotions or other vCenter related sysadmin work during the actual update.

I started from vCenter Server Appliance that I walked you through building late summer of 2012 over here:

How to install and configure vCenter Server Appliance 5.1 on ESXi 5.1, the easiest way to enable the good stuff for free, like cloning VM templates and the magic of vMotion
TinkerTry.com/free-vmware-vcenter-server-appliance-5-1-download-install-configure

I recorded video (below) of my upgrade in my small lab environment, with an otherwise idle CPU. The vCenter appliance was on an SATA2 SSD. Your results may vary. And if it's important to you to be able to roll back should the upgrade not work out, you should backup first (clone/backup/snapshot).

You also are responsible for your own lab, so you should read all about the new features and fixes:

VMware vCenter Server™ 5.1.0a Release Notes
vCenter Server™ 5.1.0a | 25 Oct 2012 | Build 880471
vCenter Server™ Appliance 5.1.0a | 25 Oct 2012 | Build 880472
Last updated: 25 Oct 2012
www.vmware.com/support/vsphere5/doc/vsphere-vcenter-server-510a-release-notes.html

Here's the steps to update your vCenter Server Appliance:

1) Login to the ESXi host with your vSphere Client
to monitor progress through the Performance or Console tab views, if you'd like:

2) Login to the appliance with your browser
where vcenter is your vCenter appliance host name or IP address:
http://_vcenter_:5480

3) Click Check Updates

4) Click Install Updates

5) Go do something else
it takes a while (15-20 minutes in my case), and is unattended (doesn't ask any questions), with web UI auto-refreshing when done

6) When it's done, you can click on Check Updates again, but you don't have to, you're really done!

Are you skeptical that it really is this easy?  Then go ahead and watch the full walk -through video:

See also:
vCenter Server Appliance 5.1 Upgrade (VCSA) Known Issues

Jan 06 2013 Update:
5.1.0.53000 Build 947940 is now out, aka, 5.1.0b, detailed at download site here:
my.vmware.com/web/vmware/details?productId=284&downloadGroup=VCL-VSP510-VC-510B-EN

and the procedure to do the upgrade is the same, seen in the walk-through video (with no audio necessary) below: