How much of existing art and science would he have been willing to sacrifice so that those who created it could instead have been working on Friendly AI?

I don't know about Eliezer, but I would be able to sacrifice quite a lot; perhaps all of art. If humanity spreads through the galaxy there will be way more than enough time for all that.

If it be replied that the work of, say, Newton or Darwin was essential in getting us to our current perspective wherein we have a hope of intelligently tackling this problem, might the same not hold true in yet unknown ways for string theorists?

It might. But their expected contribution would be much greater if they looked at the problem to see how they could contribute most effectively.

And what of Michelangelo, Beethoven, and indeed science fiction? Aren't we allowed to have similar fun today? For a living, even?

No one's saying that you're not allowed to do something. Just that it's suboptimal under their utility function, and perhaps yours.

My guess is that you overestimate how much of an altruist you are. Consider that lives can be saved using traditional methods for well under $1000. That means every time you spend $1000 on other things, your revealed preference is that having that stuff is more important to you than saving the life of another human being. If you're upset upon hearing this fact, then you're suffering from cognitive dissonance. If you're a true altruist, you'll be happy after hearing this fact, because you'll realize that you can be scoring much better on your utility function than you are currently. (Assuming for the moment that happiness corresponds with opportunities to better satisfy your utility function, which seems to be fairy common in humans.)

Regardless of whether you're a true altruist, it makes sense to spend a chunk of your time on entertainment and relaxation to spend the rest more effectively.

By the way, I would be interested to hear Eliezer address this topic in his video.