In order to be able to take advantage of the Start date time, you need to have the Traffic data enabled for the network dataset (to use Speed Profiles of TomTom/Navteq for instance).
To read more about Traffic data in Network Analyst. More info on na module in arcpy (some of which is used here).
Here is the code I use for getting the route exported to a new feature class after solving the route:
def solveroute():
#Set up the environment
arcpy.CheckOutExtension("network")
#Set up variables
networkDataset = r"C:\Program Files (x86)\ArcGIS\ArcTutor\Network Analyst\Tutorial\SanFrancisco.gdb\Transportation\Streets_ND"
stops = r"C:\GIS\Temp\test.gdb\StopsSandiego"
fastestRoute = r"C:\GIS\Temp\test.gdb\TimedRoute"
#Make a new route layer using travel time as impedance to determine fastest route
routeLayer = arcpy.na.MakeRouteLayer(networkDataset, "StopsRoute",
"Minutes",accumulate_attribute_name=["TravelTime","Meters"],start_date_time="12/30/1899").getOutput(0)
#Get the network analysis class names from the route layer
naClasses = arcpy.na.GetNAClassNames(routeLayer)
#Get the routes sublayer from the route layer
routesSublayer = arcpy.mapping.ListLayers(routeLayer, naClasses["Routes"])[0]
#Load stops
arcpy.na.AddLocations(routeLayer, naClasses["Stops"], stops)
#Solve the route layer
arcpy.na.Solve(routeLayer)
#Copy the route as a feature class
if arcpy.Exists(fastestRoute):
arcpy.Delete_management(fastestRoute)
#print int(arcpy.GetCount_management(routesSublayer).getOutput(0))
arcpy.management.CopyFeatures(routesSublayer, fastestRoute)
solveroute()
This gives me a route feature (Polyline M) saved to the disk with all the relevant attributes (including start/end time):