Are old people scared of spiders?

Anybody can be afraid of spiders, regardless of age or sex.

I'm a young woman and I'm not scared of spiders. I like them, and even have one as pet.

old people are scared of loads of things. many young people are also scared of spiders

A good many people outgrow this fear, but there are a fair share of chronies that fear them.
This fear knows no gender or age, and one strange statistic is that of people that overcome this fear, more women develop a love for spiders as opposed to men.
In light of your question, I am over 60 and grew up with LOTS of spiders about. According to my mother that had a bit of anxiety with spiders, I was playing with spiders before I could walk. So fascinated with bugs, especially spiders, I later studied them. Now one would think spotting numerous spiders daily for years, one would get used to them, take them for granted, or just plain ignore them, but my older sister has a morbid fear and hatred for spiders and refuses to change. My mother was always worried I would get bitten by a spider for playing with them, but never got a bite until I was 14 YO. (not for handling them though.)
Here are a couple of sisters with their pets:
http://www.youtube.com/watch?v=xxNpitlKc_s
http://www.youtube.com/watch?v=gxCc6E0eFpQ

Depends

I'm not old, but I'm not a young girl. Big spiders scare the **** out of me. Little ones are fine. But if it can bite, I kill it.

What do the two criteria have to do with this phobia?

Yes

yes

Deep down everybody is scared of spiders. Have you seen them? Menacing things.