> But as long as we're in sci-fi territory

That's just the thing though, there's nothing "scifi" about the idea of starlifting and extending the life of a star or storing hydrogen fuel for burning later. It's a mundane concept that could definitely be done, not a scifi idea that could not be done.

> When we're young time seems to move at a glacial pace, but the older we grow, the faster time seems to move and the more we panic about our mortality. I think a similar psychology would still play out, only over astronomical timescales.

I don't see why. A being with no mortality would also have no fear of mortality. In the future, and this part is more on the scifi scale, it is likely humanity will merge with machines and obtain essentially eternal "life" thereby, if a mechanical body could be called a form of life.

Such a being might be destroyed, but could be backed up in a thousand different places and thus able to resurrect over and over again. What would such a being have to fear?

> But at the end of the universe we'll just be staring into the dark, meaningless void. I think the second-half of the universe's lifespan could be pretty psychologically rough if a solution isn't found.

Turn inwards and build VR worlds, is the obvious answer, because such a place can be far better than the real world we live in now while feeling and looking just as real. This is considered one of the solutions to Fermi's paradox as well, in a universe where FTL is not possible, turn inwards, go virtual.