Gravitational attraction of Sun on distant object A hypothetical situation which will help me with a real-life problem. 
Several stationary objects are placed at great distances from our Sun -- say 0.1 light year, 0.5 ly, 1.0 ly. Assuming no other influences; 
What velocity would each have when it reached the Sun?
How long would each take to reach the Sun?
Assuming the object-Sun path was perpendicular to the ecliptic &amp; the Sun had its actual motion, would the objects hit the Sun, or (roughly) by how much would they miss it?
The full equation for the time for an object to drop is 
$$t =  \frac{ \arccos \Big( \sqrt{ \frac{x}{r} }\Big)  + \sqrt{ \frac{x}{r} \ ( 1 - \frac{x}{r} ) } }{ \sqrt{ 2 \mu }  }  \, r^{3/2},$$
where $x$ is the radius of the sun, $r$ is distance of the object, and $\mu=GM=1.327\times10^{20}$. (in SI units, so you will need to convert your distance to metres, and get a time in seconds.)

As $x&lt;&lt;r$ you can assume $\frac{x}{r}=0$ so the formula simplifies to 
$$\frac{\pi r^{3/2}}{2\sqrt{2\mu}}$$

For an object released at 0.1,0.5 or 1 light year that is 3000000 years, thirty million years or a hundred million years.

For the speed on impact, you can assume that the body will reach very close to the escape velocity of the sun, which is about 620 km/s. It matters little whether the body is released from 0.1, 0.5 or 1.0 lightyears, because the body will pick up speed only when it approaches the sun.

The last part of you question is unclear. If the body is at rest relative to the sun, and the various gravitational affects of other stars are ignored, then it will fall towards the sun. If on the otherhand the body is at rest relative to the centre of the galaxy, then it is moving extremely fast relative to the sun, much faster than the sun's escape velocity. It won't fall anywhere near to the sun, instead the sun will deflect it slightly as it falls towards the centre of the galaxy.