Hi. It sounds like you would want to look in Uptown and the neighborhoods surrounding Uptown like Dilworth, NoDa, Elizabeth, Plaza Midwood or Myers Park. Maybe into South End. Most neighborhoods have small parks and there are the greenways. Some zip codes are 28202, 28203, 28204, 28205, 28209.
These areas are pretty urban, have bars & restaurants, and you can find townhouses and single family homes for rent in your price range.
You won't find a lot of taxis on the street like you would in NYC, Chicago or Boston, but you can always call a taxi for a pick up. There is a reasonably decent public transportation system. Be sure to go to
CATS and do an address search from your job to the potential apartment to see the available routes and commute time.
FYI, I can empathize with you. I recently moved from NYC. I haven't owned a car in over 10 years (too hard in NYC) and I've been getting around okay so far via public transportation. I shop online and get my groceries delivered from the market (Harris Teeter) for a $4.95 delivery charge. I just put in an application for a 2 bd/2 ba townhouse with a small, fenced yard in the Uptown area and the rent is below $1500 a month.
I think you'll find what you want in Charlotte.