The joke is a pun. 'Mourning' means to feel or show deep sorrow or regret for (someone or their death), typically by following conventions such as the wearing of black clothes. Mourning sounds identical to 'morning' which means the time of day after sunrise and before noon. I suppose a heavy drinker would regret drinking so much when they are hung over the next morning.
Those who drink too much at night have to worry about the mourning after .