This could be your man.

Sulphur Rock looks like it’s within spitting distance of White River Township, see:

http://www.city-data.com/township/White-River-Independence-AR.html

…

William Prince, Private, (New) Company I, subsequently Company K, Shaver’s Regiment, Arkansas Infantry,* enlisted August 6, 1862 at Sulphur Rock by Maj. Hart for 3 years or during the war, Van Buren, Ark., restored to duty by July 1, 1863, last recorded as absent without leave since August 16/18?, 1863, no further records

Note, he must have been dropped from one of the missing Sept/Oct or Nov/Dec, 1863 muster rolls as he is not listed on the subsequently Jan/Feb, 1864 roll.

* This regiment was designated at various times as Shaver’s Regiment Arkansas Infantry, 31st Regiment Arkansas Infantry, and 38th Arkansas Infantry

M317: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Arkansas