I'll cut to the chase, because time is short. Here's what to do.
1. Check the time
I'll explain more later.
2. Contact your web host
Let them know what's going on. You want them on your side; making demands can backfire in this situation. I'd say something like:
"I'm getting lots of traffic today. Is there anything you can do to keep the website stable until this passes?"
3. Check your analytics to understand where the traffic is coming from
If you have a relationship with the source of the traffic, it can help to get in touch with them. When things went viral at the University of Oxford, the traffic was coming from another department's website - the Wildlife Conservation Research Unit. When we contacted them they explained the connection with the Jimmy Kimmel Show.
4. Establish which page the traffic is landing on
We'll come back to this later on.
5. Check the time again. How long has it been since discovery of the traffic surge?
If you work for an organisation you need to let your colleagues know at some point. This gives and also takes away. Other people can spot opportunities that come with a traffic surge. But, as other people learn of events they will come to you to ask questions. The extra interruptions can cause your practical actions to grind to a halt.
My advice is this: tell someone else, then ask them to handle communications with the rest of the organisation.
6. Optimise the user journey
If you have clear answers to steps 3 and 4, now is the time to make the most of this event. Think of the mindset of these people - what are they after? Is there anything you can promote that they'd be interested in? You could edit the page to link to that.
7. Improve the entry page
Sometimes traffic comes into a very old page. Check: is it up-to-date and accurate? I've seen traffic peaks hit pages more than 10 years old.
8. Prepare for an outage
Your website might go down at any moment. If it does, what would users see? Now is a good time to optimise that message. You can help visitors fulfil their tasks in other ways.
For example, at Oxford we provided a phone number people could call to make donations. Our website did go down, and we still received gifts.
9. Take a break
These sort of events are wild. It's useful to take a short break to let the adrenaline subside and marshal your thoughts. You'll make better decisions afterwards.
Stepping back
When you go viral, it's a shock. It happens so rarely you're not clear what to do. Hopefully this post will help you make the most of the moment.
Go easy on yourself. An event like this causes us to make hurried decisions. It's understandable if we make some mistakes.