How can one arrive at the Rasi and Natchram with the birth date and time?

Answer: Every Almanac known as panchangam gives in detail about the raminder time of a days nakshatra. When you check the same,  with the birthtime, you can arrive at the days/birth nakshatra. To know the Rasi, it is also provided in the same panchagam about which nakshatra belong to which sign. You can also avail the details of Nakshatra Padam from here.