How often do stars pass close (~1ly) to the Sun?
Here is the very study you are looking for by Bailer-Jones (2014).

Using the re-reduction of the Hipparcos astrometry, he has integrated orbits for 50,000 stars to look for objects that might come or might have come close to the Sun.

The K-dwarf Hip 85605 is the winner on that timescale, with a "90% probability of coming between 0.04 and 0.20pc between 240,000 and 470,000 years from now". 

The next best is GL710 a K-dwarf that will come within about 0.1-0.44pc in 1.3 million years.

On a statistical basis, some work has been done by Garcia Sanchez et al. (2001). They estimate, using the Hipparcos data, that encounters within 1pc occur every 2.3 million years. However, the Hipparcos data isn't complete for low-mass, faint objects. Making a correction for this, the authors estimate a $&lt;1pc$ encounter every 100,000 years. The probability for closer approaches scales as the inverse of the square of the separation. i.e. the timescales for approaches closer than 0.1pc is 100 times longer. 

There are plenty of big error bars in all this work - recall that an error of 1km/s in velocity leads to a 1pc error in position after 1 million years. All this should be nailed down really well by Gaia results in the next 2-3 years.