Complex sentence: Each player must proceed to drink the contents of a cup (typically plastic and filled with cheap beer) and upon placing the cup on a flat surface bottom-up, proceed to "tip" the cup right-side up.
Split into simpler sentences: Each player must proceed to drink the contents of a cup (typically plastic and filled with cheap beer). Then upon placing the cup on a flat surface bottom-side up, proceed to "tip" the cup right-side up.