I travelled with a budget of 15k including flight accommodation and scuba... With proper planning you can travel also with in this range. Mine was a 5 days trip...
It's by flight to port Blair. then by ship to Nicobar. Or you can take ship from Vizag. however the journey is not easy as the waters are turbulent and if you suffer from motion sickness,it will be difficult.
also it's quite difficult to go to Nicobar as there are very dangerous tribes living there. one needs to be careful.
If you are asking about days, then minimum is 5 and if you are asking about the money then it depends on where you live, still the minimum would be around 15k
Try to visit Havelock island, Radhanagar beach ( one of the best beaches in India ). While coming back you can also book helicopter ride to Port Blair. Contact me for more info..