This is actually a pretty straightforward question. The sprites don’t follow any traditions/beliefs pertaining to a menstrual cycle, because they don’t have one. Despite appearances, they are not closely related to the Great Ape family, and that’s actually one of the only types of mammal that has a cycle.
Sprites, all of my sprites, are Spirit-made, so even though they look a lot like humans there are a handful of biological differences (in the case of the wood sprites, they’re actually part plant so that right there opens up a bunch of differences).
Jacob, explaining this to Dean in Brothers Asunder:
Jacob frowned, feeling a little like a bad guy for what he was suggesting. But, as the only one good to drive the car, he knew that he had some room to insist. Dean needed the rest. Another glance down at Bowman showed that he needed it, too.
It also gave Jacob an idea. He reached out and gently scooped his hand under Bowman. The sprite flopped over on his hand before sitting up sluggishly, blinking slowly. His wings hung slack.
“What?” Bowman groused, his voice heavy with exhaustion.
Jacob looked over Bowman’s slumped form at Dean. “Bowman’s basically a plant, Dean. He’s been up and flying around since well before the sun went down. He needs to recuperate a little, at least. And you probably should, too. You had an arm busted tonight, dude.”
“Plant?” Dean asked, eyebrows scrunched in confusion. “Like energy-from-the-sun kinda plant?” He gave Bowman a slightly accusing look. “And you didn’t want to tell me that? ” he shot at the sprite, sounding put out after being lectured about how Bowman was ‘fine’ not long ago. How in the world was he going to be able to watch out for the little sprite if he didn’t even know something as important as that? At least if he or Jacob needed energy they could grab some coffee or a soda, but a plant…
“We don’t have an extra sun on hand, so you’ll need to take it easy until the morning,” Dean said tartly. “Sam won’t be happy if we run his little brother into the ground when he’s not around.” He held out his hand, beckoning Jacob to hand the sprite over. “I’ll keep an eye on small fry over here and make sure he doesn’t get into any more trouble.”