Keep in mind a few things before you go throw your hands up in the air, swear to all that is holy and start sacrificing addons on the altar. Many of your favorite addon creators have been quietly working in the background on Cataclysm versions of their addons, presumably ready to go for patch 4.0.1. As of right now, we're technically in Cataclysm, and the addon sites are marking addons that are supported in this patch as either "supports game version 4.0.1" or "Cataclysm." By making sure you update all of your addons, you will probably not have too much to worry about.
Step 1: Back up
Making a copy of your Interface and WTF folders is probably a good idea. Leave this copy off somewhere away from your main WoW directory and keep it as a "just in case" copy. Label it as such, since you can always rename the folders.
Make a list of your critical addons that you don't want to forget to update. You don't want to feel lost without that one addon you love to rely on.
Step 2: Turn on "Load out of date addons"
On the character select screen, clicking the Addons button will open up a selection window, allowing you to turn on and off installed addons. Up near the top right of the window is a button labeled "Load out of date addons." Choosing this option will load addons that are not flagged as up to date for patch 4.0.1 but will still run.
Step 3: Update your addons
Many of your favorite addons are all ready and raring to go, with updates on your favorite addon site. Addon sites like Curse and WoWInterface have a tag for compatibility with 4.0.1 as well as notifiers that let you know if an addon is ready to roll with patch 4.0.1.
Curse has a handy filter for addons tagged with its Cataclysm badge, easily pointing you to updated addons that will work with the new patch.
If you are going to be raiding on Tuesday or Wednesday, it's a good idea to get your DBM/DXE or your raid warning addon of choice up to date. Luckily, most of these addons have been ready to go and were just awaiting the patch to hit.
If you use an addon compilation, check that compilation's page for any updates, if available. You may lose some settings if you run the whole deal before updating, but if your compilation is getting an update, you should be in good shape once that update is available.
Update: Reddit user Rinascita posted an awesome list of addons that didn't have too much trouble making the jump to 4.0.1 here.
Here's the most important issue, and everyone needs to be cognizant of this danger: Addons are one of the biggest targets for hackers, account phishers and bad guys looking to take your accounts. Fake addon sites and fake addons are out there and easily mistakable from the trusted addon repositories. For your safety, only update your addons from known sources, like Curse or WoWInterface.
Stay safe, stay calm and don't worry if you have to reconfigure something now -- it's the perfect time to try new addons or a new configuration.
This is it! Patch 4.0.1 begins the slide into December's upcoming expansion. It's a whole new game from here on out. Get oriented with our roundup of everything there is to know about patch 4.0.1 and how it changes the game until the launch of World of Warcraft: Cataclysm.