Answer: Based on your description this is a fairly difficult growing site for any kind of evergreen -- most do better in full sun all day while the shade tolerant types would not appreciate hot afternoon sun from the west; evergreens also do better without the heat reflecting off the wall which can overheat in the winter as well as in the summer months and stress the plant. A needled type would do better than a broadleafed plant in this spot, especially if there is wind hitting it out of the north in the winter.
I think you might try an upright juniper such as Sky Rocket which has a naturally narrow upright shape to 3 feet wide and ultimately 15 feet tall, or you could possibly try an upright yew and keep it pruned to the shape/size you need. Another possibility if the soil is reasonably moist might be an upright columnar arborvitae.
The average time from planting to moving will depend on how much sun it receives, the soil where it is planted, and the weather from year to year. To minimize growth you would not fertilize it but instead rely on annual top dressings of compost and using an organic mulch year round in a flat layer over the root area. This will help feed the soil slowly over time as it breaks down.
Your local professionally trained nursery staff may have other suggestions based on a more detailed understanding of the growing conditions in this planting spot and your overall design goals. Good luck with your project!
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