A charge is placed at centre of circular face of cylinder of radius 'r' and length 'r' flux through remaining curved surface is (other than opposite circular face)

Cylinder is on one half side of the point charge. so, total flux through cylinder $=\frac{q}{2{ϵ}_0}$

flux through circular faced surface 

$=\frac{\mathrm{q}}{2{\mathrm{ϵ}}_0}(1-\cos \mathrm{\theta })=\frac{\mathrm{q}}{2{\mathrm{ϵ}}_0}\left(1-\frac{1}{\sqrt{2}}\right) \left(\because \mathrm{\theta }={45}^0\right)$

flux through remaining part (curved surface)

$=\frac{\mathrm{q}}{2{\mathrm{ϵ}}_0}-\frac{\mathrm{q}}{2{\mathrm{ϵ}}_0}\left(1-\frac{1}{\sqrt{2}}\right)$