Mathematica has an inverse gamma function. It is on the web page on special functions. This would suggest that the problem is at least simple enough for computer implementation.
I have just found more material on the inverse of the regularized incomplete gamma function from Mathematica. There are downloads on the site with information as well. Some of them are Mathematica notebooks and need the player which is free to be opened.
The information includes differential equations, representations through equivalent functions and series representations among other things.