In my case I don't have time to go buy another alarm clock today -- which is ridiculous anyway considering the tech in the iPhone. I'm heading to CES tomorrow morning, and I'm quite nervous about waking up on time. So I've set up an account with iPing to send me a wake-up call on my iPhone (and yeah, it's free but the message will likely have an ad -- small price to pay for convenience). I've also purchased two alarm apps -- one for my iPhone and one for my iPad. In my case I bought Clock Pro HD (US$3.99) for iPad and Awaken for my iPhone (US$0.99), which we've reviewed before. Mashable has a few recommendations as well.
A simpler solution is to set a repeating alarm instead of a one-time alarm. However, I have seen reports of repeating alarms failing as well. Some suggest you should delete all alarms and set a repeating alarm, or alternatively set a countdown timer for the hours you'd like to stay asleep -- but it appears there is no easy answer. Apple's claim that it'll start working "beginning January 3" rings untrue given my own anecdotal evidence and this report from International Business Times. It's a mess, honestly, but we're hopeful that alarms on the iPhone begin to function properly tomorrow. Or maybe Apple will issue an update addressing the problem. Or maybe we're all just waking up wrong.
Update: I managed to get a non-repeating alarm to fire off when I deleted all my previous alarms and set the new one. I'm still relying on all of my other alarm apps, just in case.