"Has achieved whatever work it was in him to do"

The number 2 formulation is not correct. The "it was" is the "to be" verb referring to the work. There needs to be such a "to be" verb in the sentence referring to the work.

It could be the work that "was there" for him to do, or that "there was " for him to do etc. You can state it as past or past perfect or future as needed.

You may think of the work as his goal or objective.

First we should say that English allows for multiple ways to say the same thing.

As mentioned @Eliot's answer, #2 is simply incorrect English.

However, #1 to me does not convey quite the same flavor as Russell's phrase does, which comes off as a bit more poetic. The phrase "whatever work was in him to do" works well enough, but "it was" to me tries—in the context given—to add a bit of wistfulness, leaving it possible to think that had the old man not grown old and tired, he'd have done more work. The "it" has left him.

But in an old man who has known human joys and sorrows, and has achieved whatever work it was in him to do, the fear of death is somewhat abject and ignoble.

I would interpret "whatever work it was in him to do" to mean his life's purpose. And "the fear of death is somewhat abject and ignoble" is saying that the fear is abject (self-abasing) and ignoble (shameful). In other words, this old man should consider the fear of death to be an unworthy emotion, one which he chooses to not have (to the extent that that's possible).

Russel goes on to say:

The best way to overcome it -- so at least it seems to me -- is to make your interests gradually wider and more impersonal, until bit by bit the walls of ego recede, and your life becomes increasingly merged in the universal life.

The "it" to be overcome here is apparently the fear of death.