Ok, if a new sun replaces our sun while were asleep we will damn sure as hell see it coming, seeing as how its a big giant ball of gas (damn hard to miss), during its trip, the second sun will damn sure as hell screw over the earth.
First thing is first, while the new star(lets call it Star II) is travelling to its destined location, the presence of two suns (one stationary and one mobile) causes many erratic things to happen to the earth. One of the more obvious results is that a good portion of oceans and rivers are vaporized and dried out, thus resulting in the massive fluctuations in water prices, effectively turning the search for water into an industry as important as the oil industry. Epic proportions of humans are effectively killed off because of a severe lack of water which is either not affordable or simply unavailable due to the fact that dual sons have vaped thier supply of water. The ecosystem is severely screwed over seeing as how fishes and other sea creatures will be killed en mass because of thier inability to cope with the temperatures causing the fishing industry to take severe hits on its economic infrastructure. Power consumption in the name of Air Conditioning are at an all time high internationally, causing severe strain to power plants, which at times may be so severe resulting in major blackouts in many important cities. Deaths continue to rise as many die by major heat strokes, dehydration, and lack of food seeing as how, many croplands will be destroyed by the heat. Fear continues to ensue, forcing many indoors and causing the dual suns issue to be the biggest political issues in the agenda. This all happens during the Sun II's trip to the old sun and will continue until the old sun flies away well out of sight and out of range. After such an event, and assuming that the two suns did not screw over our magnetosphere, this event will perplex many religious groups, cause so many to be superstitious and will vastly change our society (an obvious given, with what I outlined before)