Anthony, you are correct that the 1st Arkansas Infantry was not engaged at Richmond, Kentucky. I should have expanded my earlier remarks to state that the flag pictured in your book is that of the 1st & 15th Arkansas Regiments (Consolidated), the flag which was captured by the 14th Michigan Infantry at Jonesboro, Georgia, on September 1, 1864.
The battle honors are a combination of the 1st Arkansas and the 15th Arkansas. The 15th Arkansas was engaged at Richmond, so its battle honors for that action was included on the consolidated flag.
An interesting bit of trivia about this flag -- it was incorrectly identified by the U.S. War Department as belonging to the 1st Arkansas Artillery Regiment. Of course, there was no such outfit, but the crossed cannons, being the symbol of the U.S. artillery arm, naturally caused the Yanks some confusion. In fact, in the Confederate army the inverted (barrels down) crossed cannon was an honor accorded to a regiment which had captured an enemy battery, though many regiments (including the 1st Arkansas) mistakenly placed the honor on their flags with barrels up -- which is the insignia of the U.S. Artillery.