Those specifications are pretty tough to fill. The height, lack of width and tolerance to shade make it difficult to find a plant that can do all that. I think an arborvitae is not going to be happy in that much shade. The Carolina Cherry Laurel 'Bright N Tight' grows 8 to 10 feet tall and 6 to 8 feet wide so although it can take some shade it would be a battle keeping it narrow. If something could be done to brighten the area and get some sun in there your options would open up a bit.
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