Grafted polymers inside cylindrical tubes in good and Theta solvents are systematically studied by numerical self-consistent mean-field theory (SCFT) and molecular dynamics (MD) simulations. We demonstrate that within the SCFT and at low enough concentrations the properties of these systems are fully determined by the overall polymer concentration plus two reduced parameters which are derived from the SCFT. Via a suitable change of variables, the results for very different systems can thus map onto universal curves. We show that MD and SCFT results at finite concentrations nearly obey these mappings as well. These findings allow for generalized predictions of the polymer properties such as concentration profiles, chain end distributions, and when there are polymer-free "open" channels at the center of a cylinder. This provides useful guidance for the development of micro- and nanofluidic devices.