Obtaining extent of each polygon in shapefile using ArcPy?

Get the shape object in your cursor and access its extent property. See ArcGIS Help Working with geometry in Python:

shapeName = arcpy.Describe(inFeatures).shapeFieldName
for row in rows:
    feat = row.getValue(shapeName)
    extent = feat.extent
    print extent.XMin,extent.YMin,extent.XMax,extent.YMax

The Bounding Container toolset does exactly what you want. Should you just want code snippets, examine the functions within the scripts, one deals explicitly with extent.

EDIT

I should add that the script will add values to a Left, Right, Top and Bottom field in the created output file which can be used for subsequent processing

I just tried the Minimum Bounding Geometry (Envelope) (in Data Management) in ArcGIS 10 and it seems to do exactly the same, for all the fields.