No need to feel sorry for your reaction. While you're correct, tattoos are becoming commonplace, it's only become a normality in the past 15 years or so.
I have two full sleeves and can feel some people cringe inside when I'm interacting with them. I'm actually happy to be on the front lines of this cultural revolution, and do all that I can to show others that a tattoed person can be a bright, articulate, and valuable resource.
Thanks for sharing your honest opinion!
The weird thing is... in literally any other scenario I wouldn't give it a second thought... it's just in the office and meeting rooms. I think it's because our office is still styled from the 70s and everyone is super old and uncool... if I was in like a graphic design place or anywhere with some young blood I'd be freaked if there were no full sleevers.
People might cringe inside, but they'll forget the second that they realise you're actually awesome.