New results on the theoretical solvability of nonlinear algebraic flux correction (AFC) problems are presented and a Newton-like solution technique exploiting an efficient computation of the Jacobian is introduced. The AFC methodology is a rather new and unconventional approach to algebraically stabilize finite element discretizations of convection- dominated transport problems in a bound-preserving manner. Besides investigations concerning the theoretical solvability, the development of efficient iterative solvers seems to be one of the most challenging problems. The purpose of this paper is to take the next step to remove such obstacles: For the linear convection-reaction equation, the existence of a unique solution is shown under a mild coercivity condition and some restriction on the limiter. Additionally, the numerical effort for solving such problems is drastically reduced by the use of a highly customized implementation of the Jacobian. The benefit of this approach is illustrated by in-depth numerical studies.