A mass equivalent to, say, Saturn would cause changes detectable by our level of instrumentation even if it came in perpendicular to the plane of the elliptic. This would be whether the mass was composed of regular or "dark" matter.

The first thing noticed would be orbital changes, possibly as late as the asteroid belt but probably sooner. Should such a mass come in 90 degrees to the elliptic you would see solar mass attracted in its direction.

I'd rather it came in that way, in fact. Zip, and it's past all the planets.

But you must also realize that the solar system is very, very large and even the Sun takes up by a very small fraction of the space. I haven't done the math, but even a wandering G-type star, dark or otherwise, impacting the solar system at or beyond say, the orbit of Uranus or Neptune, wouldn't do much damage to Earth. In fact I'd kinda like to see it.