There is really great/frequent bus service connecting most of the CBD with the Hospital. I used to bus daily and only drive in for middle of the night emergencies. So unless you need to be able to get to the Hospital outside of normal hours (6AM-10PM), you can stay anywhere in the CBD and be there in 10-20 minutes. For example, if you live near Cuba and Manners you can get a 15 minute direct bus that comes every 4 minutes from about 6:30AM-10PM.
On the flip side, Newtown has its appeal in terms of being a diverse neighborhood that has a nice selection of food and services and is itself well connected by bus to everywhere. So I would not count it out.
Also, a few neighborhoods do not directly connect with the Hospital. Hataitai requires a transfer, while Miramar is direct. So even though Haitai is much closer, it takes a bit longer to get to the Hospital.