From a 45 year old perspective:
Getting older is only what you make of it. I realized long ago that one day I will die, period. I don't know when or where, but I'm 100% sure it will happen. That being said, I never had a fear of getting older.
I'm currently in what I believe to be the longest stage of my life - the working stage. I was in school for 18 years, and I've now been working for over 20, and based on my lack of retirement funds, I will be working to my grave. What I get stressed over has changed. It was girls and schoolwork as a kid, now it's money. But again, there are stresses all through life, so getting older doesn't change anything.
The only frustrating part about getting older is the body telling me to f-off. I work out every day, so I'm in "decent" shape. But, no matter, the eyesite is starting to deteriorate, the knees aren't so good, I have asthma and gout, and with any sort of exertion, my body "feels" it the next day. But I will fight my body to the end.
At 20 (just for example), it's impossible to foresee your future, so if you have a fear of the unknown, that may be a reason to be scared about getting older. But, you also have no idea what wonderful experiences you have yet to live. For me, getting married, having kids, watching them grow up, buying neat toys, travelling, etc - why be scared of growing old at 20 with some knowledge that there are a lot of great things such as the things I've expereienced to look forward to, even though at 20 you have no idea what they are.
I take each day in stride. This may sound a little wacked, but I've always considered life one big game, and I'm a willing player. I know the ground rules - I live on earth, I was born, I will die. Everything else is up to me. I'm part of the rat race like most of us are. I wish I had found another way, but alas I ended up being a typical working person to make money to buy things. No cash winfalls, no inventions to make me rich quick - just the basic 9 to 5 work hard ethic.
In hindsight, I say don't be scared. Enjoy each day. In truth, you have no idea how many tomorrows there are.