Servos usually use a simple protocol based on pulses. Servo boards are designed to generate these pattern(s). DC-Motors are usually driven using a H-bridge (wikipedia), which uses a (constant) controlled current flow. So technically it is possible, but not really practical to use a servo board to drive a DC motor.
Before looking at motors, you need to define a few parameters, apart from the load mass and leadscrew pitch. First, what is the motion profile? I assume the device won't be in constant motion. Is there a defined sequence of movements? What distance and elapsed time for each move? What elapsed time between moves?
Next, the load. Is it affected by gravity? Are there other external forces acting on the load? What about friction? You should also know the diameter and material of the leadscrew (for inertia calculation).
With all of that, you can proceed to calculate peak torque, RMS torque, and speed required at the drive end of the lead screw. That will indicate what sort of motor you need.
Then you need to consider the ratio of the reflected load inertia to the motor's rotor inertia. A high ratio will necessitate de-tuning the position loop in order to get stability. Ideally, the ratio is 1:1 (hard to achieve), but under 3:1 is generally very good. Under 6:1 is generally acceptable. If the coupling is "stiff", 10:1 can be OK.
If the inertia ratio needs reduction, first look at a bigger motor. Otherwise, a gearbox may be required. Reflected inertia reduces as the square of the gear ratio. But speed requirements increase.
Best Answer
In general, for a motor with integrated gearbox, specifications in the data sheet are "at the shaft" which is the shaft that comes out of the gearbox.
In the case of hobby RC servos, like the one you're linking to, it is always the case that the torque takes into account the gear reduction (because it's a bigger number, thus better marketing.)
When you say "lift 4-5 lbs" you do not specify the distance of the lever arm. For a winch that winds a cable, that arm is the distance from center-of-horn to outermost-layer-of-cable. If you attach an arm, it's distance from center of servo horn to center of weight, projected to the ground plane.
You also don't specify how far you need to lift the load.
And you should know that most motors (and certainly all cheaper, hobby-style servos) are specified in maximum "stall" torque, which is not a level of performance they can provide on a continuous cycle. If you stall the motor for more than a few seconds, you are likely to burn it out. The "holding torque" or "continuous torque" is typically 0.3 to 0.15 times the stall torque, depending on motor construction, cooling, etc. This important data isn't even generally available for hobby servos, but is often specified for industrial automation components.
Also note that that servo likely has, at most, a 180 degree rotation angle.
A better Stack Exchange to ask this question on might be the Robotics Stack Exchange. I don't know how to go about moving questions/answers, though.