Let's see...

For a coordinate patch try . Now we can calculate the normal at the point , .

For , we have

if my memory is not failing me again . For the first one say, we have

, and so on.

The principal curvatures are just the eigenvalues of .

For the Gaussian and mean curvature, either use the known formulae that include the principal curvatures, or play smart and notice that the cyhlinder and the plane are isometric, so...