Can i develop application in .net to perform specific functionalities of arcGis like interpolation etc.
Yes, you can. ARCGIS does provide tools that be used in winform application. But, first of all make sure that your organization will purchase license for ARCGIS. It is not free and quite expensive.
how much time will be required for intermediate level developer
This involves GIS skills along with application development skills. And as none of us know your skill set, therefore we cannot tell that.