How to determine the mass of a runaway star? I would like to know how do astronomers estimate the mass of a star that has been ejected from its galaxy.  What kind of accuracy can be achieved?
OK, I'll assume you are talking about this star or some other high-mass "runaway star".

Most massive stars are born in rich clusters and are often part of multiple system. Is not unusual for such stars to get ejected from these regions are high velocities, either as a result of dynamical interactions in multiple systems or when they are companions to stars that explode as supernovae.

The steps to estimate the mass are: (i) Obtain very detailed optical and UV spectroscopy. From this you estimate the spectral type of the star, it's effective temperature and its (current) mass-loss rate. (ii) You combine this with a known distance (the star in the link was in the Large Magellanic Cloud, so the distance was known fairly accurately) to estimate a luminosity. (iii) You compare the luminosity, temperature and mass-loss rate with the predictions of a specialised high-mass stellar evolutionary model, which includes mass-loss and rotation. The comparison yields an estimate of the mass and age of the star.

The details are found in Evans et al. (2010). It is probably fair to say that the estimate of $90\ M_{\odot}$, is uncertain by a few tens of solar masses.