For the partner: in the operating room
Nine in ten partners are glad they were there. One in two were anxious the whole time.
In a prospective study of 100 partners present at planned and unplanned cesareans:
Both halves of that are true, and only the first one usually gets said. The single factor associated with lower partner satisfaction was the baby being taken to intensive care.
In a smaller controlled study of 64 women having a planned cesarean, those accompanied by their partner reported lower anxiety than those who were not. It was not randomized and it was done at one centre, so read it as a direction rather than a measurement.
Babies held skin to skin by their father cried less, settled sooner and reached a drowsy state earlier. Heart rate and temperature were more stable. In the largest of the four, 108 newborns, feeding behaviour came earlier and breastfeeding lasted longer, and the fathers themselves scored lower for anxiety and depression.
That trial reported only "P<.05" for every one of its outcomes, with no effect sizes and no named instruments. Both of those findings therefore rest on a single small study reported without the numbers that would let you judge them.
If the mother is unable to provide SSC immediately after birth, the father-infant SSC is a valuable alternative.
Partner skin to skin is a replacement for the baby being alone in a cot. It is not a replacement for the mother. If she can, she should, and your turn comes after.
Worth arranging before the day rather than negotiating during it: whether you can be present if the anesthetic has to be converted to a general, where you should stand, whether you can take photographs, and who goes with the baby if the baby has to leave the room. Those are policy questions with different answers in different hospitals, and the time to ask is at a prenatal visit.