The hospital should have breast pumps that you can use if you don't want to put bub to the breast. You could then use the sippy spoon to feed as it won't be enough to warrant a bottle usually. I know it's encouraged to at least give the baby colostrum if you're not wanting to breastfeed so good on you for considering that. Stimulation of the breast does encourage the milk to come in but it's also hormonal, so even if you don't express the colostrum you may find yourself engorged.

For me personally and a few others I have spoken too, the milk coming in for the first baby is often worse. I was quite surprised actually that I was no where near as engorged as the first time. I hope that you have an easier time with it this time around