Large-scale Monte Carlo simulations are presented for a coarse-grained model of cylindrical molecular brushes adsorbed on a flat structureless substrate, varying both the chain length N of the side chains and the backbone chain length N-b. For the case of good solvent conditions, both the cases of weak adsorption (only 10 to 15% of the monomers being bound to the surface) and strong adsorption (similar to 40% of the monomers being bound to the surface, forcing the bottle brush into an almost 2D conformation) are studied. We focus on the scaling of the total linear dimensions of the cylindrical brush with both chain lengths N and N-b, demonstrating a crossover from rod-like behavior (for not very large N-b) to the scaling of 2D self-avoiding walks. Despite the fact that snapshot pictures suggest a "worm-like" picture as a coarse-grained description of such cylindrical brushes, the Kratky-Porod worm-like chain model fails because there is no regime where Gaussian statistics applies. We compare the stiffness (orientational correlations of backbone bonds, persistence length estimates, etc.) of the adsorbed bottle brush polymers with their corresponding 3D nonadsorbed counterparts. Consequences for the discussion of pertinent experiments are briefly discussed.