Pula is a long way from Split - Aer Lingus to Dubrovnik would be closer.
You'd likely have to take a bus from Pula to Split - probably an overnight bus (trains in Croatia aren't very useful)
Jadrolinija as quinno mentions are the (state owned?) Croatian Ferry company - they certainly have frequent ferries from Split to Brac - it's only a couple of miles, I believe the ferry port on Brac is called Bol.
Jadrolinija do run a ferry down the coast too which would go overnight from Rijeka which isn't too far from Pula down to Split - but the bus is probably faster and more comfortable (there's a new motorway down a good stretch of the coast now which makes it much faster)