The second one, U2, is done between 3 and 10 days, so if you stay in the hospital for the full three days, you'll often be able to have U2 done in the hospital—unless you're born on a Wednesday morning: they don't have a pædiatrician of their own, but get daily visits from the Mariahilf hospital but only on weekdays. So if a child is born on Wednesday or Thursday, it won't be old enough for U2 by Friday but you'll usually be discharged by the weekend. Then you'll have to arrange for the U2 to be done at a pædiatrician's.
U3 is done between 4 and 6 weeks, and includes more of the same: blood tests, seeing whether the child is growing and gaining weight, testing reflexes, etc.
We also talked about newborn's jaundice, which may require you to stay at the hospital depending on the bilirubin levels and how well it responds to phototherapy. Apparently, babies who have a different blood group from their mother (if the mother has type-0 blood) or who have a different Rhesus factor are more at risk from this, and the bilirubin level which indicates phototherapy is lower for them. Though that shouldn't be a problem for us, fortunately; as far as I know, we're both 0+.
At the end, we all went to have another look at the newborns' station, and saw the phototherapy lamp in operation (which is pretty glaring, so the children get a kind of cloth spectacles to protect their eyes), then we went another floor down to look at the delivery rooms and the bathtub, which Stella and I had already seen twice :)