There's a small airport in Norwich with a few flights, but none to Dublin, or any other Irish airport, any longer (I think Aer Arran used to fly there for a while)
There are flights to Amsterdam, Edinburgh and Manchester from Norwich so I guess in theory you could connect through those places .. but it's likely to be time consuming and expensive.
So flying to London and going from there would make most sense ... definitely fly to Stansted - that's the closest London Airport to Norwich
From Stansted you can get a train via Cambridge to Norwich that is reasonably efficient ... There may be buses as well, check the National Express website