Can anyone suggest me to Extracting Single City datasets or feature dataset from master data base by one click using Python (arcpy).
In Arcpy and python there is no single one-click solution unless you have prefilled out an attribute and/or spatial query to select the feature and copy it to your desired location. You can examine the arcpy object methods in the link and the tools in arctoolbox. If you have a workflow you are trying to simulate, then you need to provide some specifics as to what steps you are trying to remove or simplify
Sounds like you want to build an Add-In. What is a Python add-in?—Help | ArcGIS Desktop
You can import a Python script and control functions based on user mouse clicks or keyboard commands.