We investigate the formation of charged patterns on the surface of cylindrical micelles from co-assembled cationic and anionic amphiphiles. The competition between the net incompatibility chi (which arises from the different chemical nature of oppositely charged molecules) and electrostatic interactions (which prevent macroscopic segregation) results in the formation of surface domains. We employ Monte Carlo simulations to study the domains at thermal equilibrium. Our results extend previous work by studying the effect of the Bjerrum length l(B) at different values of the cylinder's radius R and chi and analyze how it affects the transition between helical, ring, and isotropic patterns. A critical surface in the space (l(B), R, chi) separating these three phases is found, and we show how it corresponds to a first-order phase transition. This confirms that the Bjerrum length l(B) is a significant parameter in the control of the helical-ring transition; the ring pattern is strongly associated with short-range forces, whereas the helical pattern develops from dominant long-range electrostatic interactions.