A fox youkai gains 3 tails after 100 years worth of training, 6 tails after 1000 years. When it reaches 9 tails, it's blessed a great power and become a strongest fox youkai. (No one knows how long it takes)
A 9-tails fox is considered one of the strongest youkai in Japan, most cunning youkai in China and most evil in Vietnam.
So as ZUN stated Ran has 9-tails, I believe she has >1500 years of training. Maybe older than Yukari, even.
And how it dies... it's quite vague, since most 9-tails fox in myth had to be killed by a hero before it destroy everything. The fox has 10 lives contains in its 9 tails and main body. If you kill by destroying the body, it will turn to a flame and resurrect. Rumor has it, when a Kyuubi dies by age, it'll turn back to a normal fox.
The age of a youkai. I believe aside of Gods, everything can die from aging. Many youkai spent their life searching how to be young/immortal. Yukari may used her gapping ability to restrict the time inside her body, while Yuka/Byakuren has so much magic power that automatically maintains their appearance for a long time. Aya is a tengu so yeah. Tewi... she is the luckiest xD
I thought fox youkai where at their most powerful state when they first gain 9 tails after 800/1000 years, and then lose them again over the course of hundreds of years 'till they have four again, turning them into Tenko: nigh Godlike and immortal beings. Maybe only a select few of Kyubi can turn into a Tenko after several reincarnations.
But alot of characters are explicitly stated to be immortal. While most youkai are indeed just long-lived, Byakuren/Greater Magicians and Eirin/Lunarians are unable to die of old age. Celestials and Immortals are described as perfectly immortal (though Celestials have to fight shinigami and Immortals aren't as durable as most other Touhou races, getting blown up with a few shots).
But as far as I see, that the characters that are unable to die by natural means (as in, they can't die of age or sickness like mortals, and have to be forcefully killed/are unable to die at all) are:
Mokou & Kaguya (Immortal)
Eirin, Kotohime and Yorihime (Never ending life)
Byakuren (Ageless)
Remilia & Flandre (Undead)
Yuyuko, Mima, Murasa & the Prismrivers (Dead. Youmu doesn't count since she's still half human and could be killed.)
Kanako, Suwako, Hina, Minoriko and Shizuha (Faith Gods, and are thus immortal. Will vanish when they lose all their faith though)
Shinki, Shikieiki and Dragon God (True Gods/Infinite Beings, and are thus perfectly immortal. Might be able to be destroyed by a being greater than them, but there's probably hardly anything stronger than them)
Komachi (Death itself)
Tenshi (Perfectly immortal thanks to Heavenly Peaches. Shinigami still chase them down however)
Yukari might be immortal, thanks to her ability, but it isn't explicitly stated. Yuuka might have once been a human, so that would mean she's a Greater Magician now, and be thus ageless.