We've been asked several times on here about transportation between Oxford and Memphis and no one seems to have a good answer. You might try some of the places listed on the links below. I can't speak as to the reliability of any of them as I've never used one. We'd love for you to come back after you use one and give a bit of a report for fellow travelers.
Thank you! I will give them a call and post what i find out
You're so welcome.
The University of Mississippi's Outreach Airport Shuttle service is great. I live and work in Oxford and use the service frequently.