There aren't many buildings downtown that are strictly apartments. Most units available for rent are condos that people rent out as investment properties.
Greensborough Court is all apartment, I believe, and in a really good location.
Distinctive Apartment Dwelling at The Lofts at Greensborough Court
You can find good deals on second and third floor walkups over ground floor retail. But most only advertise in the classifieds or on craigslist.
1br - Downtown GSO
If you're really interested, I would call Downtown Greensboro, Inc., which is a non-profit that supports growth and development downtown. There number is 336-379-0060, and someone there should be able to provide you with up-to-date availability listings.
Good luck! And welcome!
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Relocating to Greensboro, NC, would like a place downtown..
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