I know it's possible to freeze gnocchi before cooking it, but is it possible to cook it fully the day before? Maybe store in the fridge and then reheat in a saute pan with butter the next day? Please advise, and thanks!
Absolutely. Par-cook it, drain, toss it with some oil, cool in a single layer and refrigerate. Take it out of the fridge 30 minutes or so before sautéing.
Yes, and whenever I make gnocchi, I make at least a double batch, and toss them with a little oil, then cool the extras and freeze them in heavy duty ziploc bags in the freezer. So handy to pull out and either microwave, from frozen, or heat in a frying pan with a little butter and whatever else tastes good . . . onions, bacon, sundried tomatoes, pesto ...mmmm.
Most certainly you can. In fact, you're describing what restaurants do as a matter of routine. Before storing in the refrigerator, be sure to toss it in some olive oil to prevent it from sticking together. Just enough to gently coat it is fine.