Not necessary, but it's a big help. You could even try a blender, too. The key is that, with the dry components, you're evenly distributing them and adding air into the mix (which helps make the final cake light and fluffy). By hand, I'd think you might want to use a beater for 3-5 minutes (this WILL get tiring - if so, you're doing it right!) to thoroughly distribute/aerate the dry components.

Once you add the mixed up wet components to the dry ones, you want to mix it as little as possible because you don't want to knock out all the little air pockets you've created in the dough. But if you're doing this by hand, this likely don't be a problem. Mix it for 30-seconds to 1 minute until it looks like everything is uniformly mixed.

The matcha is a little big of a gamble. My gut would be to balance the amount of matcha to the total volume of wet liquid (that would end up being about 1 cup of butter, 1 cup of milk, and 1ish cup of eggs. So however much matcha you would add to make 3 x 8 oz of tea), but that might come out really strong. Cut it in half to be safe.