When you start with a Python module, there are several solutions to find the available functions. One of them is `dir`:

    geom = feat.GetGeometryRef()
    print dir(geom)
    ['AddGeometry', 'AddGeometryDirectly', 'AddPoint', 'AddPoint_2D', 'Area', 'AssignSpatialReference', 'Boundary', 'Buffer', 'Centroid', 'Clone', 'CloseRings', 'Contains', 'ConvexHull', 'Crosses', 'Destroy', 'Difference', 'Disjoint', 'Distance', 'Empty', 'Equal', 'Equals', 'ExportToGML', 'ExportToJson', 'ExportToKML', 'ExportToWkb', 'ExportToWkt', 'FlattenTo2D', 'GetArea', 'GetBoundary', 'GetCoordinateDimension', 'GetDimension', 'GetEnvelope', 'GetEnvelope3D', 'GetGeometryCount', 'GetGeometryName', 'GetGeometryRef', 'GetGeometryType', 'GetPoint', 'GetPointCount', 'GetPoint_2D', 'GetPoints', 'GetSpatialReference', 'GetX', 'GetY', 'GetZ', 'Intersect', 'Intersection', 'Intersects', 'IsEmpty', 'IsRing', 'IsSimple', 'IsValid', 'Length', 'Overlaps', 'PointOnSurface', 'Segmentize', 'SetCoordinateDimension', 'SetPoint', 'SetPoint_2D', 'Simplify', 'SimplifyPreserveTopology', 'SymDifference', 'SymmetricDifference', 'Touches', 'Transform', 'TransformTo', 'Union', 'UnionCascaded', 'Within', 'WkbSize', '__class__', '__del__', '__delattr__', '__dict__', '__doc__', '__format__', '__getattr__', '__getattribute__', '__hash__', '__init__', '__iter__', '__module__', '__new__', '__reduce__', '__reduce_ex__', '__repr__', '__setattr__', '__setstate__', '__sizeof__', '__str__', '__subclasshook__', '__swig_destroy__', '__swig_getmethods__', '__swig_setmethods__', '__weakref__', 'next', 'this']


and no `getNumInteriorRings`: apparently this method is not exposed in the python bindings.  
But if you look at [[gdal-dev] Polygon topology][1]:

> In fact you have to use the Geometry.GetGeometryCount() that
returns 1 (the exterior ring) + the number of interior rings. So
polygon.GetGeometryCount() - 1 should return the number of interior rings

So a polygon only:

    geom.GetGeometryCount() 
    1

For a polygon whith one hole:


![enter image description here][2]

    geom.GetGeometryCount()
    2 # -> one hole

and 

    geom.GetGeometryRef(0) # -> exterior ring = LinearRing, in red
    geom.GetGeometryRef(1) # -> interior ring = LinearRing

![enter image description here][3]

    exterior = Geometry(ogr.wkbPolygon)
    exterior.AddGeometry(geom.GetGeometryRef(0))
![enter image description here][4]

    interior = Geometry(ogr.wkbPolygon)
    exterior.AddGeometry(geom.GetGeometryRef(1))

![enter image description here][5]



But it is easier with [fiona][6] (also based on GDAL/OGR)

    import fiona
    layer = fiona.open("myshape.shp")
    feat = layer.next()
    print feat
    {'geometry': {'type': 'Polygon', 'coordinates': [[(1149490.1097279801, 691044.60910800309), (1191579.10975257, 691044.60910800309), (1191579.10975257, 648030.57611583965), (1149490.1097279801, 648030.57611583965), (1149490.1097279801, 691044.60910800309)], [(1154115.2745658469, 686419.44427013607), (1154115.2745658469, 653118.25743749342), (1165678.186660514, 653118.25743749342), (1165678.186660514, 686419.44427013607), (1154115.2745658469, 686419.44427013607)]]}, 'type': 'Feature', 'id': '0', 'properties': OrderedDict([(u'FID', 0.0)])}
    print feat['geometry']['type'])
    Polygon
    print len(feat['geometry']['coordinates']
    2
    # outer ring
    print feat['geometry']['coordinates'][0]
    [(1149490.1097279801, 691044.60910800309), (1191579.10975257, 691044.60910800309), (1191579.10975257, 648030.57611583965), (1149490.1097279801, 648030.57611583965), (1149490.1097279801, 691044.60910800309)]
    # inner ring
    print feat['geometry']['coordinates'][7]
    [(1154115.2745658469, 686419.44427013607), (1154115.2745658469, 653118.25743749342), (1165678.186660514, 653118.25743749342), (1165678.186660514, 686419.44427013607), (1154115.2745658469, 686419.44427013607)]
>>> 
     


  [1]: http://osgeo-org.1560.x6.nabble.com/gdal-dev-Polygon-topology-td3745761.html
  [2]: https://i.sstatic.net/qm9h1.jpg
  [3]: https://i.sstatic.net/vg0xc.jpg
  [4]: https://i.sstatic.net/EfIIc.jpg
  [5]: https://i.sstatic.net/LAYWp.jpg
  [6]: http://toblerity.org/fiona/manual.html
  [7]: http://osgeo-org.1560.x6.nabble.com/gdal-dev-Polygon-topology-td3745761.html