We present a rigorous semi-analytical formulation for the analysis of electromagnetic (EM) scattering from a cylinder constructed from a metasurface represented using surface susceptibilities. The formulation uses the Generalized Sheet Transition Conditions (GSTCs) to represent the surface and eigenfunction expansion (EFE) of the incident and scattered fields, by exploiting their angular periodicity. Incorporating a completely general non-uniform surface formulation of 36 susceptibility components, a matrix equation is formulated that can be solved for the field harmonic coefficients. The paper illustrates the methodology with a number of examples; including two formed from a finite sized practical unit-cell exhibiting a normally oriented magnetic resonance which is compared with commercial EM full-wave solver; other examples present surfaces that have a modulated gain/loss profile (i.e. amplitude modulation) and polarization conversion. It is found that for all cases the EFE solution very accurately captures the scattered fields in both the interior and exterior regions of the surface. Detailed convergence studies are further presented including the effect of susceptibility modulation on the number of terms needed in the EFEs, to reach a correct field solution. The proposed EFE framework is further compared with a second GSTC based simulator using an Integral Equations (IE) approach and it is found that the IE-GSTC simulated fields approach that of the EFE as the surface discretization is increased. The proposed EFE approach is thus a quick, rigorous methodology which, although limited in the geometry which it can model, has many advantages for investigation into the use of GSTCs, application development and providing a baseline for simulation studies.