Just as the title says: How old was Aragog when he died? Do we know if he died of old age or injury/illness?
Aragog lives from 1942 until April of 1997, roughly 55 years. From the Harry Potter Wiki:
In 1996, Aragog contracted an unknown illness over the summer, and despite Hagrid's attempts to heal and comfort Aragog by feeding him giant grubs, he eventually died on 20 April, 1997.
Considering that Aragog's wife (Mosag) had already died I suspect the lifespan of Acromantula is not excessive. Aragog has already outlived his wife and become blind by the time Harry Potter meets him. Although he died of an illness, I would figure he would have died of old age sooner rather than later.
Most normal spiders live for one year or less. Some tarantulas (of which I believe Aragog resembles) may live as long as 20 years.
He died of illness at the age of (approximately) 54-57.
When was Aragog born?
In Deathly Hallows we see the graves of James and Lily Potter, which state:
DIED 31 OCTOBER 1981
In Philosopher's Stone Hagrid says that this occurred 10 years earlier when Harry was one year old:
All anyone knows is, he turned up in the village where you was all living, on Halloween ten years ago. You was just a year old.
This would mean that the Philosopher's Stone school year was 1991-1992. In Chamber of Secrets Harry finds Tom Riddle's diary at the beginning of 1993. This can be derived as follows:
They used the Polyjuice Potion on Christmas:
No one, not even someone dreading taking Polyjuice Potion later, could fail to enjoy Christmas dinner at Hogwarts.
Hermione's transformation didn't go as planned and she was in the hospital wing for several weeks:
Hermione remained in the hospital wing for several weeks.
On the way from visiting Hermione in the hospital wing Harry and Ron heard Filch's outburst:
Ron was just saying saying he wished he had asked Hermione how many rat tails you were supposed to add to a Hair-Raising Potion when an angry outburst from the floor above reached their ears.
They follow the sounds and end up in Moaning Myrtle's bathroom where they find the diary:
Harry and Ron looked under the sink where Myrtle was pointing. A small, thin book lay there.
So several weeks into 1993 Harry acquires the diary. The diary is dated 50 years previously:
Harry saw at once that it was a diary, and the faded year on the cover told him it was fifty years old.
Thus, the diary was for the year 1943.
Harry later enters the diary on the page for June 13th:
The pages of the diary began to blow as though caught in a high wind, stopping halfway through the month of June. Mouth hanging open, Harry saw that the little square for June thirteenth seemed to have turned into a minuscule television screen. His hands trembling slightly, he raised the book to press his eye against the little window, and before he knew what was happening, he was tilting forward; the window was widening, he felt his body leave his bed, and he was pitched headfirst through the opening in the page, into a whirl of color and shadow.
In the diary memory we meet Aragog for the first time:
A vast, low-slung, hairy body and a tangle of black legs; a gleam of many eyes and a pair of razor-sharp pincers — Riddle raised his wand again, but he was too late.
From this we know that Aragog must have been born before June 13th of 1943.
When Harry and Ron later meet Aragog in the Forbidden Forest, he tells them that he was not born in the castle:
"I!" said Aragog, clicking angrily. "I was not born in the castle. I come from a distant land. A traveler gave me to Hagrid when I was an egg. Hagrid was only a boy but he cared for me, hidden in a cupboard in the castle, feeding me on scraps from the table.
We don't know exactly when Hagrid received the egg, though. We know that Hagrid was expelled in his third year, as he tells this to Harry in Philosopher's Stone:
"Oh, well — I was at Hogwarts meself but I — er — got expelled, ter tell yeh the truth. In me third year.
It is possible that Hagrid only got Aragog that year, but it is also possible that he already had Aragog in his first two years at Hogwarts. He probably did not have Aragog for any significant amount of time before starting at Hogwarts in his first year, because Aragog only mentions being raised in the castle.
Additionally, after their confrontation Harry seems to have assumed that Aragog was hatched in the castle:
Ron gave a loud snort. Evidently, hatching Aragog in a cupboard wasn't his idea of being innocent.
Fantastic Beasts and Where to Find Them tells us how long it takes acromantula eggs to hatch:
The young hatch in six to eight weeks.
So this puts Aragog's birth at approximately somewhere between the latter part of 1940 (when Hagrid started at Hogwarts) and the first half of 1943 (when he was definitely already hidden in the castle).
When did Aragog die?
In Half-Blood Prince there was a sign mentioning the date of the Apparition test:
There had been a certain amount of excitement earlier when they had come back from dinner to find a new sign on the notice board that announced the date for their Apparition Test. Those who would be seventeen on or before the first test date, the twenty-first of April, had the option of signing up for additional practice sessions, which would take place (heavily supervised) in Hogsmeade.
On the day of the Apparition Test, the trio was sitting in the courtyard:
They were sitting with Ron in a sunny corner of the courtyard after lundh. Hermione and Ron were both clutching a Ministry of Magic leaflet — Common Apparition Mistakes and How to Avoid Them — for they were taking their tests that very afternoon, but by and large the leaflets had not proved soothing to the nerves.
While there, a girl delivered a letter from Hagrid:
Ron gave a start and tried to hide behind Hermione as a girl came around the corner.
"It isn't Lavender," said Hermione wearily.
"Oh, good," said Ron, relaxing.
"Harry Potter?" said the girl. "I was asked to give you this."
The letter from Hagrid said:
Aragog died last night.
Thus, Aragog died the night before April 21st, i.e. April 20th. Based on the aforementioned years, this must have been 1997.
Therefore, Aragog lived from somewhere between between the second half of 1940 and the first half of 1943 to April 20th 1997, for a total of about 54 to almost 57 years.
How did Aragog die?
In the beginning of the term in Half-Blood Prince Hagrid tells the protagonists that Aragog is sick and is dying:
"It's... Aragog.... I think he's dyin'.... He got ill over the summer an' he's not gettin' better.... I don' know what I'll do if he... if he... We've bin tergether so long...."
Later in the year he tells them that Aragog's condition is deteriorating:
"Aragog's worse, I bin readin' to him — didn' get up ter dinner till jus' now an' then Professor Sprout told me abou' Ron! How bad is he?"
And eventually Aragog succumbed to his illness, as mentioned, on April 21st, 1997.