Shakespeare had one cherry eye pop at 4 months, and the next one 3 weeks after we got the first one fixed! After lots of research and consults, we opted to have them both removed. Everyone I know that had it tucked, had it pop out and ultimately had to have it removed. The vet explained that they only get 2% of their tears from this area so even in the remote chance that it is damaged and they develop dry eye, they would have most likely had dry eye anyways.
He did not put him under, simply gave him (something) to calm and relax him. He was done and ready to leave in less than 15 minutes (we opted to stay longer to observe him since we live 45 minutes from the vet). All they do is clamp the cherry eye and cut it off. We had no bleeding, crusting, any issues at all with either eye. He wore a cone for a few days just to protect it but it didn't bother him at all.