I grew up in the Memphis area, so I'll take a shot at this. First, the public golf there isn't anything special, so don't get your hopes up too high!
Most of the better public courses are outside the city, and much of it is to the south in Mississippi.
Kirkwood probably stands out as one of the nicer public jobs, but it is worth gathering some intelligence first because it seems to go between phases of different owners and upkeep. It is in Holly Springs, MS - about 45 minutes south of the city.
Other decent courses in northern MS that might be worth a look (closer to Memphis) - Cherokee Valley, Plantation, Wedgewood.
North of the city you have Quail Ridge in Bartlett. It has a few quirky holes, but overall I like the course and it is usually in pretty good shape. Big Creek and the renovated Navy Course (I can't remember what the new name is) aren't terrible options if you are in that part of the map.
As mentioned here already, you may want to consider Tunica (45-60 minutes south of the city) which is the second biggest gaming area in the country (bigger than Atlantic City in gaming area - at least it was at one point not sure with some of AC's growth.) There are several casino options there and some higher end golf courses (higher end price - not sure about quality since I haven't seen them firsthand.) The Grand Casino runs one of the more popular courses in the area and there is another one that I have read good reviews about called Tunica National.
Let me know if you want anything more specific.