A lattice model for a symmetrical copolymer "bottle brush" molecule, where two types (A, B) of flexible side chains are grafted with one chain end to a rigid backbone, is studied by a variant of the pruned-enriched Rosenbluth method (PERM), allowing for simultaneous growth of all side chains in the Monte Carlo sampling. Choosing repulsive binary interactions between unlike monomers and varying the solvent quality, it is found that phase separation into an A-rich part of the cylindrical molecule and a B-rich part can occur only locally. Long-range order (in the direction of the backbone) does not occur, and hence the transition from the randomly mixed state of the bottle brush to the phase separated structure is strongly rounded, in contrast to corresponding mean-field predictions. This lack of a phase transition can be understood from an analogy with spin models in one space dimension. We predict that the range of microphase separation along the bottle brush backbone can be controlled on the nanoscale by varying the solvent quality.