I have a multi-ring space station built around an asteroid, but not physically connected. Station has significantly more mass than the asteroid. As the two approach a star and swing around it, will the station need to do anything to maintain the same relative position and trajectory to the asteroid? Will they gain the same velocity?
If the two bodies are gravitationally bound, they will follow the same trajectory as long as the tidal forces do not exceed the gravitational binding.
Just look at our planet and its moon: they have different masses but they keep following the same orbit around the Sun without any adjustment maneuver.
Your concern shouldn't be about orbiting the star - that's the easy part. You just need the centre of mass of the station-asteroid system to be at the right place moving with the right velocity.
Your concern should be about the dynamics of the station-asteroid system itself.
A ring with a body at its centre is in an unstable equilibrium, and even small perturbations will lead to the two bodies colliding. The gravitational potential of the asteroid will increase towards the centre of the ring, meaning it will want to fall towards the ring where it will collide with it. The symmetry prevents it from doing so if set up perfectly, but a passing micrometeorite will modify the potential enough that it is now slightly off the peak of the potential and will inevitably fall off it into the ring.
This is easiest to show for a ring, but similar results apply to almost any sort of enclosing structure.
A perfect sphere does not suffer from this (it has a completely flat potential within it), but maintaining a perfectly isometric distribution of mass as people move around and live there daily lives would be next to impossible.
What you're going to need are a lot of station-keeping engines attached to the station that keep the asteroid in the proper place relative to the station.
Essentially, this works the same way as an acrobat balancing a pole on their forehead. Making smaller adjustments more often takes much less energy than making large adjustment less frequently. If done well, it shouldn't have any impact on the overall orbit, just as the acrobat can take whatever path they choose whilst keeping the pole vertical.
Ships arriving and departing may make things a little trickier, so station-keeping may require good information on the mass of all ships as they arrive and leave, as well as good monitoring of their velocity so it can properly account for whatever momentum they impart when they dock (or take as they leave).
If the asteroid is orbiting the station, this isn't an issue and you're basically all clear.
Note: I have not discussed tidal forces. If the asteroid and station are orbiting each other the station-keeping necessary to keep the asteroid on the gravitational potential peak will handle that just fine; if the asteroid is orbiting the station you'd need tidal forces greater than the attraction of the asteroid to the station for it to pose a problem, at which point you don't have an asteroid orbiting the station anyway, just two bodies orbiting the star in very close orbits.
It's complicated. The issue is, any tiny drift of the asteroid relative to the ring will tend to grow.
The path a finite sized object takes in an orbit is affected by the details of its shape. These effects are somewhat smaller than the general scale of the tides. If the mass of the asteroid is not spherical, for example, it will want to follow a slightly different path. This is because the parts that are farther away from the star want to lag behind the parts that are closer.
Tides will do weird things also. The space station is based on rings. When the rings are edge-on to the star, the tide acts parallel to the plane of the ring. But as you orbit past the star it may be a challenge to keep that true. If the ring is in the same plane as the orbit you may be able avoid having tides flip the ring around. Otherwise, as you pass the star the tides will want to flip the rings around.
If the asteroid is at all "egg shaped," meaning it has one axis longer than the others, it will have a preferred direction under the effects of tides. As you orbit past the star this will change. The asteroid will tend to swing around to point at the star. As it does it will tend to have very small changes in its orbit. If it manages to get even a little off center then tides will tend to grow that larger and larger.
So what you want is to shape the asteroid to be as nearly spherical as possible. And orient the ring to be in the same plane as the orbit you are following. And you want to make sure the asteroid is as perfectly centered as possible.
Then some tiny little correcting jets would probably still be required to adjust for the effects of solar wind. The rings probably feel a different level of that to the asteroid. Even a minute difference may off-center the asteroid. And the orbit past the star will be months giving this tiny force a long time to act. Talking about a tiny little thruster about the strength of an aerosol can.