AFAIK its about 8 miles.
The [broken link removed]bus does not have terminus any longer at Eden Quay and the closest it appears to go to the Hospital is to the Square.
It would be a better option to get the [broken link removed] from Abbey St. (off-peak times) or Connolly Station (terminus - peak times). It passes in front of the hospital and would be quicker than getting stuck in the traffic especially on the Greenhills Road.