Air Taxi Captain reported while preparing to take off at a non-towered airport; another aircraft took off from the same runway opposite end without making any radio calls. Captain held position on runway until other aircraft had passed.

Narrative

On Day 0; Captain and First Officer were operating Aircraft X from Jonesboro; AR to ZZZ. At XA53 crew and passengers were boarded and after all checks were completed; I made a CTAF call on 123.00 that Aircraft X was taxing to Runway 23 via Hotel; Alpha and we taxied to Runway 23 without incident. At the hold short lines for Runway 23; and after final takeoff checks were completed; I checked to make sure there was not an aircraft on final or on the runway and my First Officer also verified that both the final and the runway were clear. Not seeing any other aircraft and hearing no radio calls; I proceeded onto Runway 23 for departure. I taxied onto Runway 23 and as we were about to take off; we saw some sort of tail dragger pop into the air on Runway 5 headed toward us. We held our position until the small aircraft was clear of the runway environment and then took off without further incident. At no time on the ground or while we were in the pattern did we hear this aircraft make any sort of calls up until the time we switched over to Memphis Center.