As a first approximation, it's just Newtonian physics: The force between two objects with mass M and m at distance r is F = GMm/r^(2) where G is the gravitational constant. If you calculate the acceleration of the object in the field of a different mass, then the mass of the object cancels: a = F/m = GM/r^(2). That means we don't need to know the comet mass (which can be difficult to estimate) to find its acceleration. Measure its position a few times to find its location and velocity and you can calculate its future trajectory step by step. If only the Sun's gravity is relevant then the comet will fly in an ellipse and you can use Kepler's laws. See if the trajectory intersects Earth's orbit, and see if both objects will be in the same place at the same time.

To refine the orbit estimates, you want to look at non-gravitational effects: Is the object ejecting gas because the Sun heats it up? Is there significant radiation pressure? Anything else? You can also use some corrections from relativistic effects compared to Newtonian physics. All that goes into better estimates how the object will accelerate, so it will improve future position estimates.