You could try having a look at www.toandfromtheairport.com (Galway)to see what it has to say about getting to Dublin from Galway by taxi, bus etc.
I see you have had no replies to your query. I think a taxi would be your best option. The hospital is on the west side too so the journey shouldn't take more than 10-15 minutes. I have no idea whet the cost would be.