If only William IV and Queen Adelaide could've seen into the future in 1828, eh?
We also had the workhouse, later the Brighton General, the Women's hopital, the QA children's hospital in Dyke Road and the isolation hospital in Bevendean.
Modern medicine eh?
The 7 stops outside Brighton station (comes down from Seven Dials) and outside the RSCH. They are very frequent (every 7 minutes) so train and bus would be my choice.
There are two buses from Brighton to the Princess Royal, 272 and 271, free of charge if you show your appointment letter.