Apologies for some of the vagueness, but I recall that this was a short story I read sometime around 2000/2001, though I suspect it was published well before that.
The premise is that whenever someone dies, they go to this sort of purgatory. There, they meet the person who died just before them, who tells them what to expect. That person then goes on to the afterlife. The person who has just died is then responsible for waiting until the next person dies somewhere in the world, at which point they pass on the info, and so on.
At some point in the distant future, mankind has achieved a sort of immortality by backing themselves up to massive servers. Because no one is dying anymore, the last man to die has been waiting in purgatory, alone, for a very long time.
There's a brief power outage, causing a hiccup in the immortality system. A young woman, who suddenly finds herself temporarily dead, appears in purgatory and talks with the man. Eventually, the power is reactivated and she disappears, returning to "life", and the man is left by himself.