Say a Justice is barely hanging on to her life. Can a president nominate and the Senate confirm a Justice in anticipation of their death? Do they have to wait until the person dies? Can't it be a conditional (conditional on joining the court in the event of the death) pick dependent upon the Justice passing? Seems to be a time saver.
Can a President nominate a US Supreme Court Justice in anticipation of a sitting Justice's demise?
First, Supreme Court Justices can only be removed by Death, Impeachment & Removal, or by Resigning. Therefore, as long as an ill Justice hasn't resigned and hasn't died, he or she is still part of the SCOTUS.
There is nothing in the constitution that speaks about replacement - temporarily or permanent - either.
As a result, if the "replacement" justice is formally nominated before the demise, technically he or she won't be a replacement but an additional member.
The number of Justices in the SCOTUS is not specified by the Constitution. However, it is determined by a law. See this answer.
So without changing the law, it is not possible to add a 10th Justice.
Summarizing the above:
No, it's not possible to formally nominate and confirm a replacement Justice before the current one has died or resigned.
Of course, the president can publicly name someone that he intends to nominate as soon as the current justice demises. And the Senate is free to discuss anything, including this person.
But after the actual death, there will still be a formal nomination and a formal confirmation. And the president and senate are no way bound to do whatever they announced. So I don't think this will be done in practice.
That said, every president has a short-list just in case a justice unexpectedly dies. This short-list is the closest thing to what you are looking for at the moment.
As @hszmv pointed out in a comment, Trump released a list of candidates he was considering for the nomination on the Supreme Court as part of his 2016 campaign and of the two seats he has replaced, both candidates were on this list.
(I couldn't find a link to this list. A quick internet search yielded mostly the lists provided by many lobbying organizations. If anyone has a link to the original list, feel free to add it in and remove this note)