Cut off crown and peel from pineapple, then quarter pineapple lengthwise and core. Cut fruit into 1-in. chunks and whirl in a blender, a portion at a time, with 2 cups water until puréed. Pour purée through a fine strainer into a large bowl, squeezing pulp to remove all juice; discard pulp. (If using jarred juice, mix with 2 cups water in a bowl.)
Whirl ginger in blender with 2 more cups water until puréed. Pour through strainer into pineapple juice, squeezing liquid from pulp; discard pulp. Stir in lemon juice, then add sugar to taste.
Pour into a pitcher, cover, and chill until cold, at least 2 hours. Stir, then pour over ice.
Make ahead: Up to 3 days, chilled.