Do I need to be a prime member to use Fulfillment by Amazon?

So I bought a textbook last night, said it had fulfillment by Amazon. During checkout it said Prime members get free two day shipping, but I'm not a prime member so I selected the $4.99 shipping which arrives this Friday, but the Prime message didn't go away. It said "become a Prime member to get fast $4.99 shipping!" I clicked Checkout anyway, and the transaction went through.

Today, the business day is almost over and I've had no notification that my item has shipped. Should I be worried? Are they just slow? Will they ever even ship it?