Your 2004 Civic - which trim is it? I'm trying to figure out what engine yours actually is. To my knowledge, 2004 was the 7th generation and all trims except the Si came with a 1.7 D series single cam. The Si had a 2.0L K series dual cam. From your picture, it really looks like the back of my K series.
In my car, I personally installed the oil cooler from a K series that had it. All said and done it cost maybe $120 in parts, and I got a new water pump out of the deal while I was at it. Maybe it would help with the coolant circulation, by giving the coolant both an inlet and an outlet?
I wouldn't try to bore the hole out.
