You bring up a bunch of issues, so this answer has multiple parts.
Could an astronaut in a near-future space ship survive transit through our asteroid belt?
The image of a field full of rock is an out and out myth. The Asteroid Belt is mostly empty space.
It is 4% of the Moon's mass spread over 50 trillion trillion cubic km. Of that mass, most is locked up within only a handful of objects. About one third is accounted for by Ceres, while Vesta, Pallas, and Hygiea make up 9%, 7%, and 3% (respectively). That leaves just under half of the Asteroid Belt's mass (1.9% of the Moon's mass) for the rest of that 50 trillion trillion cubic km.
There are only an estimated 800 trillion asteroids larger than a metre within the Belt. That may sound like a lot at first blush, but 800 trillion asteroids divided by 50 trillion trillion cubic km is just under one asteroid per 50 billion cubic km.
Worrying about hitting one of those while traveling through the Belt would be akin to driving over salt flats the area of the United States with only 2000 people scattered across them and worrying about hitting someone.
You could well say "what about smaller objects?", but the situation doesn't change very much. 4% of the Moon's mass over 50 trillion trillion cubic km is only 60 mg per cubic km (60 picograms per cubic metre). There is no worst case distribution where there'll always be something dangerous to hit, and since most of that mass is actually concentrated in large objects, which themselves tend to be concentrated in several orbital groups, it's extremely easy to not hit anything in the Asteroid Belt.
This shouldn't be shocking. If there was any significant concentration of mass scattered throughout the Belt, it would quickly collapse into a planet. That's why much of what little mass is there collapsed into Ceres.
TL;DR: In fiction, ships traveling through the Belt must swerve around asteroids. In reality, the many space probes we've sent through the Belt paid minimal attention to collision risks. This won't change much even with higher speed travel.
In the near future we may find cargo ships fleeing space pirates!
Thruster technology is such that the ship can successfully maneuver through Sol's asteroid belt at an average velocity of 1,000 Km/s.
Keep in mind that ships traveling at a constant velocity, by definition, aren't using their engines/thrusters. In space, you don't need to push to stay in motion. You only need to push to speed up and to slow down.
A single Merlin engine (currently used on SpaceX rockets) can already generate nearly 1000 kN of thrust. That means 8 or 9 modern engines could accelerate 30 metric tons (30000 kg) to 1000 km/s if fuel wasn't the problem that it currently is. Unless people in your near future setting are still struggling with fuel mass and power sources like we are today, there's no reason to imagine any ship in your universe would need more than minutes to get up to speed. In other words, the vast majority of their travel time should be unpowered.
This raises a problem for pirates. How do they detect ships with no drive plumes in a region (literally) many trillions of times larger than the volume of the Earth? Hell, even if the ships ran hot for their entire journeys, spotting them in such a large zone would still be next to impossible without high powered ground-based tracking.
- If pirates can see cargo ships over these distances, why wouldn't the cargo ships be able to see those pirate ships. Remember, they'll have to burn towards the cargo ship after spotting it. That would give the cargo ship plenty of warning (and time to start increasing their own speed). I mean it's not like pirate ships can just sit next to "shipping lanes" and wait. (There's no sitting in space. There's only falling around the Sun, planets, moons, etc.) The only way to stay in a lane is to actually travel along its orbital path. What's worse, the paths cargo ships must take change as the planets orbit the Sun. Even worse than that, the lane for any given orbital configuration is highly speed dependent. That means one ship traveling at 950 km/s would have a vastly different "shipping lane" from a ship traveling at 1300 km/s. So, there's no easy way for a pirate to travel along a shipping lane at a slower pace and wait for other ships to catch up.
If a ship requires no power to stay at 1000 km/s, of course it can maneuver at that speed. You have to define what kind of course correction constitutes "successful". Success is only a matter of warning time and a ship's ability to accelerate. If you're saying that these ships are able to detect obstacles with enough lead time to briefly use their engines for course corrections, then you've already answered your own question.
Question: Could our astronaut survive two hours of space flight (a run of 7,200,000 Km or 180X the circumference of the Earth) inside the asteroid belt without dying from the maneuvers necessary to avoid impacts?
Given that the odds of needing to maneuver are nearly non-existent, the astronaut wouldn't have to worry. However, even when maneuvers are needed, unless you've somehow accidentally ended up right on top of a planet, minor nudges are all that will be required.
After two hours the pirates make a mistake, take an impact dead center of the windshield, and due to the force of explosive decompression, find themselves hurtling deeper into the belt like little human torpedoes. Our hero can slow down and avoid all future impacts.
Maybe you want to set this in Saturn's rings? That is the only location in the Solar System with the kind of debris density you're imagining. Of course, you'd have to find a reason for two ships to want to stay inside the rings for so long.
I'm sorry. I don't mean to rip holes throughout your entire idea. Things in space just don't work like things on Earth. If you want input based on the facts, than the fact is that most traditional forms of conflict don't work in space.
- Fuel dictates travel speed, not travel distance.
- The speeds you can get up to and slow down from determine what destinations you can travel to.
- The visitability of destinations (for your speed capabilities) depends on orbit and mass.
- Space is big and bumping into people or things is hard.
- If you are close enough to see other people, everyone can see everyone else.
- Running after ships (or running away) is like shooting a gun. You can have a significant effect in the short term, but you can't do it for very long. In the case of space ships, you'll run out of fuel.
- Etc, etc.