In The Chronicles Of Amber series, Corwin states that he lived for 300+ years on Earth, and Oberon, his father, lived for thousands of years. How long do the Amber nobility live? Do they die of old age?
Poetically described as 'centuries without end', but not truly endless on an infinite time scale. Long enough that the life of a man equates to but a 'brief span' (and without factoring in the whole time differential between worlds)
As I stood on a hilltop and the evening began around me, it seemed as if I looked out over every camp I had ever stood within, stretching on and on over the miles and the centuries without end. I suddenly felt tears come into my eyes, for the men who are not like the lords of Amber, living but a brief span and passing into dust, that so many of them must meet their ends upon the battlefields of the world.
Nine Princes in Amber - Chapter 6 (p. 127-128)
They do however seem to age, as they are born, they have a childhood, and they grow old. Though they will not 'die in bed', (of old age), as Corwin jests at the possibility
While I had often said that I wanted to die in bed, what I really meant was that in my old age I wanted to be stepped on by an elephant while making love.
Guns of Avalon - Chapter 6 (p. 136)
Benedict, the eldest Prince of Amber, has been alive and well for several millennia, and is yet still in his prime.
"I fear Benedict. He is the Master of Arms for Amber. Can you conceive of a millennium? A thousand years? Several of them? Can you understand a man who, for almost every day of a lifetime like that, has spent some time dwelling with weapons, tactics, strategy?
- The Great Book of Amber - (p. 204)
The lifespan of a Prince appears to be potentially limitless, as they are essentially immortal. Practically however, they are not invulnerable to being slain, and since a not insignificant portion of them will die from plots, assasination, or battle before reaching their 5,000th year, their life expectancy is on the order of "several millennia" to "multiple ages of the world" the latter/upper end defined by Oberon and Dworkin as Kings.
I think TimSparrow summed it up best, by referring to Tolkien and his description of elves:
"(they) die not till the world dies, unless they are slain or waste in grief (and to both these seeming deaths they are subject).”