If you had health education lessons in your high-school, you must be very familiar with the importance of first aid box. Don’t let a small cut be the cause of your death because of an infection or blood poisoning.
Your immune system will not start working like that of the superwoman because obviously, you are not from Krypton. You are from Earth (which is about to end by the way) and you have no superpowers. Your body is still vulnerable to flu, viruses, and other diseases so it is better to carry painkillers along with you.
If you suffer from some other diseases like asthma, high blood pressure, diabetes etc, make sure that you do have with you all your routine medicines.