The poem is about what drinking does to time. In the first image, a clock loses its face, so it can still run but can't tell you anything. The speaker says the same thing about themselves: after 8 o'clock, "I have no when." Past a certain hour, the evening stops being measured and the speaker loses track of where they are in the night.
The second stanza turns the clock into a "drunken pair" that "tocks," hand in hand. The word plays on "tick-tock," but only half of it survives, as if the clock has slurred its own sound. The "hands" are both a clock's hands and two people holding on to each other, perhaps for balance, perhaps for company. Given the title, "Happy hour," the mood is loose and playful, with a slight unease underneath. The fun is real, but so is the sense of drifting without anything to mark time.