There are a number of philosophies on how one should approach this. Some would suggest disabling your addons and waiting a few days until the storm subsides. Some have even said you should clear out your Addons folder, or your WTF folder. Clearing out the Addons folder gets rid of the addons, while deleting your WTF folder will clear all player and character settings. I personally go this route when I find that most of my addons are acting "buggy" or weird, or when I want to start with a clean slate.
I recently wrote an article on how you can automate the updating process, which I would recommend you take a look at, the servers are down anyway! Alright, really I would recommend reading through the comments of that post, as some of your fellow readers have some great advice and insight you may find helpful. Also, the folks over at WoWInterface contacted me in regards to their own addon updater, which looks promising as well. They have been busy testing it and have released a Windows version as well as a Mac/Linux version, both of which merit a look in my opinion. WoW Interface also released their Patch 2.4 Survival Guide, which I profiled recently. All this just screams that you've got options, so keep reading!
There are a couple of things you can do to make this process even easier. First of all, select "Load out-of-date Addons" in your Addons interface. You can find out more about installing and enabling addons in my article about addon basics. This can help you determine which of your addons absolutely need to be disabled and updated. This can, however, lead to some serious in-game issues, so I can only advise you to go this route if you're fairly familiar with your addons and how to identify which of them are having issues.
For many of the more popular addons, the developers have been hard at work on 2.4 updates and will have fresh, working versions available for download. The developer of Omen has had a 2.4 compliant version in development for weeks now, so you can bet other good developer's are also hard at work. This is where the trouble with new content patches and updating your addons is found. Many of your favorite addon sites will be bogged down or even possibly offline if they can't handle the sudden influx in traffic. Some sites offer a low-bandwidth version, which can help prevent traffic jams.
If you can wait a couple of days without your addons, this can be a way to dodge the bullet. Think of it like staying in your seat during the credits of a film so that the people can dissipate, saving you the trouble of dealing with the mobs. I sometimes use macros to help replace some of the functionality of my addons. It's a very limited kind of help, but it also gives me a reason to test new macros and learn more about them.
Try to keep your cool and just ride this wave as it comes into shore, you'll be up and running again. Regardless of how you feel about buffs or nerfs to your class, you can bet that your addons will take a beating this patch, so just hunker down and weather the storm, my friends, for this too shall pass.