postprocessing GPS coordinates with RINEX data to improve precision?

To perform post-processing you typically need a GPS that can record raw pseudorange and carrier cycle data. Very few consumer level GPS support this functionality, and those that do require software modification. I am not sure about android, most likely it would depend on the GPS chip that the phone is using.

Trimble produces a GPS Analyst extension for ArcGIS which can perform post-processing with RINEX data. On the open source side, it looks like GPSTk can be used.