I would say for the most part OSU fans (the ones outside of Greater Cincinnati), UC was irrelevant prior to the past couple of seasons. I moved to Columbus in 2001. I remember wearing a polo shirt with a C-paw a few years ago and a co-worker asking me what that was. When I told her it was UC, she said she didn't even realize we had a football team (she was being serious--- and this is a woman who is a big football fan and whose husband played for OSU and in the NFL).
The folks up here had a lot of respect for the MAC schools though, particularly Miami, Ohio U. and Bowling Green. They did not command the same respect as a B10 school, but in their eye it was a good alternative if a kid was not good enough to play in the B10 and they saw the academics as top shelf. Even in 2003 when Miami had their nice season they did not get all upset about, it was another opportunity for them to gloat on a national audience the strength of Ohio football.
UC's run the last few years was different. I would say even before the t-shirts they were less comfortable at UC's success. I have always maintained OSU was happy for the MAC schools because they knew that deep down any success they had would be short term and at the end of the day, those people were OSU fans at heart too. I think when it comes to UC, they realize the loyalty to OSU is not the same. That bothers OSU fans. When you throw in the t-shirts, the WLW jingles and whatnot, it makes it intolerable for them. OSU fans don't like to be disrespected, particularly by people they believe are supposed to be fans of their school.
When I moved up here in 2001, my eyes were opened as to how people up here view the city of Cincinnati as a whole. I can tell you they don't like it. Not in the manner that they speak with pride about other parts of Ohio-- Columbus, Cleveland, Akron, Youngtown, Massilon, etc. They perceive Cincinnati as an extension of Kentucky, not one of their own. To be fair, the city of Cincinnati sees itself differently too, maybe that is what bothers them. Go to all of these other towns I mentioned and the highlight of their life is likely some momment they went somewhere in Columbus or Cleveland. In Cincinnati, we all know people who have never left their side of town, let alone venture up to Cleveland.