More specifically, would an astronaut in a modern space suit be able to survive in such an environment for more than a few seconds?
Oh that poor astronaut.
In addition to the other excellent answers about the state of the universe (or lack of a state) that far in the future, there's also the problem of The Big Rip.
The expansion of the universe is accelerating. This means the space between everything is getting bigger like the surface of an expanding balloon, except that surface is 3 dimensional space. If it keeps accelerating, as it appears to be, eventually it will be accelerating so fast to overcome the fundamental forces that hold things together.
First, gravity will be overcome. Galaxies will fall apart. Then solar systems. Then stars and planets and other bodies held together by gravity will disintegrate.
Then electromagnetic bonds between molecules will be overcome and complex structures, such as people, will be torn apart. Then the molecules themselves will be ripped apart into atoms. Then their electrons will be lost.
Eventually the acceleration will become so fast the strong nuclear force will not be able to hold atoms together. Then protons and neutrons will be pulled apart into quarks...
You get the idea. Ultimately, if the acceleration continues, the universe will be expanding faster than the speed of light. (This is ok because "the speed of light" is actually the maximum speed of information transfer between two points in space, space itself has no such limitation). At this point no information can be transferred between two points in space, they're moving apart too fast. The universe is dead.
What does this mean for our time traveling astronaut? They're immediately torn into a cloud of fundamental particles expanding in all directions faster then the speed of light. Good epitaph. Same thing for any instruments you send.
That is all according to our current understanding and much of it is hypothetical. That understanding has changed in the last few decades due to the "discovery" of Dark Energy and Dark Matter which make up about 95% of the energy in the universe.
Dark Matter we're sure exists, we can see its gravitational effects, but we don't know what it is, but we do have some candidates. On the other hand, Dark Energy we have no idea what it might be. "Dark Energy" is really a placeholder for "the universe is expanding too fast and we don't know why".
In reality the answer is right now we're not confident about what the distant universe will look like because we don't know what about 95% of the universe is.