I don't know how to calculate toque in this kind of vertical setup.
All information i found on the web always lead me to the torque
calculation for motor driving wheels(which is simply the tangential
force multiply the moment arm).
The torque is the tangential force multiplied by the moment arm, no matter what the orientation.
Say you are trying to spin up something like a 0.5 kg bicycle wheel that is 1 m in diameter.
At 1 rpm, the tread has a linear velocity of pir(1 revolution/60 seconds) =~= 0.026 m/s.
To get this up to speed from a standing start in, say, 1/60 of a second requires an acceleration of every particle of the rim of a = velocity/time =~= (0.026 ms/s)/( 1/60 second ) =~= 1.6 m/s^2.
Since practically all of the mass of a bicycle wheel is at the rim, the moment arm of that 1 m diameter wheel is 0.5 m. (Other kinds of objects have more mass nearer the center; for the same total weight and diameter, they have lower moment of inertia and so require less torque).
So, as always, F = ma, so in this case F = (0.5 kg)(1.6 m/s^2) = 0.8 kg*m/2^2 = 0.8 N.
So, as always, the torque is the tangential force multiplied by the moment arm, or in this case t = rF = (0.5 m)(0.8 N) = 0.4 Nm = 40 Ncm.
The actual torque required would be somewhat more to overcome static friction, dynamic friction, etc.
If you were willing to gradually bring it up to speed over a longer time, then you would require less torque.
(Lots of standard NEMA 17 motors can easily supply 20 Ncm of torque).
Once it is up to speed, the motor only needs to supply enough torque to overcome the dynamic friction in order to maintain that speed.
Motors are generally designed to supply torque, not handle heavy radial-thrust loads.
If the stuff you want to turn is unbalanced,
spinning it at high speeds causes radial forces ("centrifugal force", aka "unbalanced washing-machine effect").
As OricTosh pointed out,
this has no effect on the torque required from the motor or the energy required to turn it;
but you may need to either
(a) add more weight on the other side to balance it, or
(b) add a bearing or two to support those radial forces, or
to prevent those loads from pushing the motor rotor "sideways" enough to cause damage.
Motors are generally designed to supply torque, not to handle heavy axial-thrust loads.
If you have a heavy load on a vertical axis, you might need to
support that load with a
(perhaps on the "other" end of a double-shaft motor --
Single-shaft vs Double-shaft motors
to prevent the load from pushing the axle through the motor enough to cause damage.