Is expiation (kaffarah) obligatory for someone who swore not to do something for the rest of his life, then made an exception saying "a few months," especially if it is difficult for him to adhere to it?

The exception must be connected to the oath for it to benefit the swearer. So, if you intended by your oath to abstain from doing something for the rest of your life, specifying this matter after the oath is complete will not be beneficial. However, if doing this thing is better than leaving it, then you may do it and offer expiation for the oath. This expiation is feeding ten poor people, or clothing them, or fasting three days if you cannot find that, according to the saying of the Prophet, peace and blessings be upon him: "Whoever swears an oath and then sees something better than it, let him do what is better and expiate for his oath." It is advisable to be forbearing and avoid anger.