Average October temperatures and climate in Costa Rica at this time of the year are at an agreeable 60-85 degrees Fahrenheit. Humidity is pegged at 85% while average rainfall rises to 15 inches from the usual 10.In October, a popular province to be at is Puerto Limon.he height of the rainy season in Costa Rica is during the month of October usually more then in September. Here, the rains come in the afternoons and sometimes even more early into the night. Rainy Season in this country however, is.