Yes it's OK to put Neosporin ointment inside your nose but don't put it too far into your nose.
Just put a little bit in your nose and it can help heal noses and nostrils that have irritation or rashes from drying out and it can help keep the nostrils moist.
The Neosporin is also supposed to help keep the flu virus and cold virus from entering your nose which can help some during cold and flu season.
But the cold and flu virus can still get into your body and infect you through your mouth when you touch your mouth and face.
So always wash your hands more often when you have the cold or flu and wash your hands after blowing your nose or coughing into your hands.