I performed this upgrade on a couple of vCenter 5.1 systems recently. Just install 5.1 U1a over the top of 5.1. The environment I upgraded used a multi-site SSO configuration so I upgraded each of the elements one at a time, i.e. SSO, Inventory Service, vCenter Server, Web Client and Update Manager. If you have a simple Install then you should be able to run the Simple Install from the CD to upgrade.
I think I needed to provide the admin@SYSTEM-DOMAIN password to perform the upgrade so make sure you know what this is.
Also, just to make sure, I would recommend that you have a backup of the server and the DBs (vCenter, SSO and Update Manager).
What I noticed was that I did not have to upgrade my vSphere Client as the 5.1 client worked with the 5.1U1a server, although I did upgrade the client.