Broadening the perspective a little bit, remember that the province of Ulster comprises three counties from the Republic as well as the six of Northern Ireland:
By Future Perfect at Sunrisederivative work: Hogweard (talk) - Ireland_complete.svg, Public Domain,
https://commons.wikimedia.org/w/index.php?curid=16459639
I think that's just asking to be cycled around. I did this in October 1985, starting and finishing in Dublin, and going clockwise. This was my route, about 520 miles in total, probably the most satisfyingly complete tour I ever did:
It's opinion, of course, but in my view the four "must sees" are the Mourne coast road in South Down, the Fermanagh lakeland, County Donegal, and the Antrim Coast. They all link up easily. The particular locations mentioned in this thread are mostly in Antrim and would be hard to miss when following the coast all the way round.
If I ever go again, I'd particularly like to see the twin cathedrals in Armagh, both dedicated to St Patrick, symbolically opposing each other across a valley running through the city. At the time I was advised against venturing that way.