Shochu is a spirit from Japan that is comprised of rice, barley or sweet potato with a light alcohol content. Just wanted to clarify, felt like I was being a bit douchey just saying "Shochu" like everyone knows what the f it is. I just found out what the hell it is recently. It drinks like Vodka with the faint character of Sake(the rice anyway).
The flavor profile of Shochu from Japan is all over the map, and there are some pretty strikingly unusual varieties. The Kuma shochu is made from rice and tastes a lot like Chinese liquor. My favorite is the type made from sweet potatoes, usually from Kagoshima prefecture and southern Miyazaki prefecture. Good luck and enjoy it!
Actually shochu can be made from one of any of thirty base ingredients. Sometimes a blend of two. Usually this would be rice and Imo (sweet potato.) Some of the other base ingredients besides rice, barley or imo are brown sugar, dates, carrots, soba, chestnuts, sake lees, and just about any grain that is grown in Japan.
While usually the strength is is bottled at is 25% abv /50 proof, it can be bottled at up to 42% / 84 proof, and I have heard about even higher strength bottlings recently.
I think a light rice shochu would make a nice summery cocktail with watermelon and yuzu like you mention. With shochu cocktails you need to use a higher proportion of shochu as the base than with higher proof spirits so as to get the delicate shochu flavor to come through. I'd use 2.5-3 oz of the shochu, 1-2 oz watermelon juice, 1/4-1/2 oz yuzu (so little because it is very tart and has a lot of flavor and more would overwhelm the drink.) and either a 1/2 oz of simple syrup or turbinado syrup, or 1/2 oz of a liqueur.
If you used a barley shochu in this recipe I would definately use either turbinado syrup or Benedictine or Drambuie as the sweetener.
with an imo shochu you need to match the deep and earthy flavor and I might not use the yuzu but instead 1/4 - 1/2 oz of Carpano Antica sweet vermouth or a nicely oxidized sherry.
You could then shake on ice and serve up, or shake then pour into a tall glass filled with ice and top with seltzer for a summer cooler.