In a resealable plastic bag, toss the apples with the lime zest, lime juice and 2 tsp. water. Freeze until solid, about 8 hours.
In a small saucepan, bring the sugar and 2 cups water to a boil. Lower the heat and simmer until the sugar is dissolved, about 5 minutes; let cool.
Transfer half of the sugar syrup to a blender. Working with a handful of frozen apples at a time and continuing until half of the apples have been added, puree the mixture. Strain and press through a fine-mesh strainer set over a bowl. Repeat with the remaining syrup and apples.
Immediately divide the mixture among the molds and freeze until partially frozen, about 1 hour. Insert the cinnamon sticks and freeze until solid, about 5 hours. Run warm water over the molds to remove the pops.
