When you work a 9-5 at an established company there are usually several degrees of separation between you and the anxiety that comes from a business. The sales and support teams will be handling the clients. The testers will handle bug reporting. The management will worry about biz dev and making money.
At a typical startup you're going to be taking the brunt of all of those worries. The stakes are higher because you're usually fighting for the company's survival, not just your Christmas bonus check. For these reasons and more I'd say working under pressure is an extremely important skill if you're looking to get into startups.
Startups aren't for everyone. If you're a brilliant programmer who doesn't want to be pressured into quickly adapting, perhaps you should apply to Google instead.
The current fad with startups seems to be to operate under extreme pressure, so you should expect to find that as a requirement at most of them. But, I bet they don't all work that way, so keep looking and good luck. :-)