Fun fact, Spirit and Shadow aren’t the only temples in that game you can do out of order. You can do the Fire Temple first, as there is only one chest that requires arrows and it doesn’t have anything crazy important in it. Water Temple can also be cleared without arrows, but you do have to get a bit more creative with some non-intuitive jumps and either saving and resetting or using a warp song to go back to the dungeon entrance without raising the water level in order to do that. Spirit Temple does require arrows but nothing else from the other dungeons is required so you could do it as early as your second dungeon. Even then I think you can abuse invincibility frames to skip an eye switch and I can’t remember anything else that required arrows in there. Shadow Temple is the most restrictive, but only because you have to clear Forest, Fire, and Water in order to gain access to it. Once you’re actually in the dungeon you’ll find that arrows are the only item from the other dungeons you actually have to use to beat it.