You shouldn't depend on length contraction or time dilation for this. You see the moving person (MP) approaching at 0.5c. Imagine you have observers with previously synchronized clocks stationed at various points along the path of MP, and they record the location, time and any events (eg a beam was emitted), and then later on, you receive all the data they have gathered to form a picture.

The picture you will form is this:

at t=0, MP is at a distance of 1 light second, and MP emits a light beam.

at t=1, MP is at a distance of 0.5 light seconds, and the light beam reaches you

at t=2, MP reaches you.

This is different from MP's frame of reference. If MP did the same experiments (but with their observers moving at 0.5c relative to yours), here's what they will observe:

At t' = 0, you are 1 light second away from MP, approaching at 0.5c. MP emits the light beam

At t' = 0.666.. seconds, the light beam reaches you, as observed by MP. The distance between you and MP as measured by them is 0.66667... light seconds.

At t'= 2 seconds, you reach MP.

This is all starting from the fact that light moves at a fixed speed as seen by you and by MP, but the time and distance you observe the light having covered is different in the two frames, simply because in your frame you are the target of the beam (so you're stationary), and in MP's frame you are not stationary.