How do you become faster at walking? You can walk for half your life and still be a slow walker. You learn to walk faster by increasing the pace at which you walk.
How do you become a faster driver? You can drive for 20 years and still be a slow driver. In fact you might even drive more slowly than you used to. You learn to drive faster by driving at a faster pace.
With anything, you push yourself well beyond comfortable limits. You don't work longer hours on these. You do short, intense, very uncomfortable hours. I'd say one very difficult programming task a day is worth more than doing a year of simple scripting.
Learning to think on difficult questions faster is the exact same thing. You do it by pushing yourself. Push yourself harder and once it's as hard as it gets, push yourself to go faster.
If you have a choice on picking difficult tasks, go for the hard ones that you're not sure you can solve. You can only improve yourself by being uncomfortable. If that costs too much, train. Tackle hard problems outside the office. Professional sports players train when they're not on call, why shouldn't the rest of us do it?
I have a fear of answering a question incorrectly, so I tend to get nervous which has an impact on my performance, especially at interviews and in meetings.
What is the worst that can happen? Will they fire you? Delay your raise/promotion? Look down on you?
You need the feedback to improve. Good managers love employees who are willing to make mistakes as they're the ones who improve fastest.
As with everything else, the best way to avoid being nervous is to force yourself to go through situations that make you nervous. If things go wrong, you will still be nervous, but know how to handle the pressure. If things go right, you'll lose the fear. Win-win either way.