Yeah, the domain age matters! However since its about buying an old domain you should know that domain history also matters. Beware that the domain that you buy must not have any bad history/reputation.
Older domain is the good to the web site ranking, i.e. it is an additional factor to that; but other SEO works are the primary factor of that.
If you can get a hold of a relevant, clean, backlinked domain, jump on it. Otherwise, try not to inherit other people's problems. It is probably available for a reason.
It matters a little bit. The older the domain, the more "established" you are. What also helps if you bought the domain for several years, so it doesn't expire for several years. This shows that your domain is more dedicated to the website, instead of a website that might disappear in a year.