How can we determine during a lunar eclipse whether the earth moves faster or the moon using minimum instruments?
At the moment of the lunar eclipse both the Earth and Moon are moving tangentially to the line joining them to the Sun, and their velocities are parallel.
The diagram above shows the moon just before, during and just after the lunar eclipse. I'm guessing the question is asking whether the velocity of the Moon, $V_m$, is greater or less than the velocity of the Earth, $V_e$.
If so, then you just have to see what direction the Earth's shadow moves across the Moon. If $V_m > V_e$ then the Moon starts at the lower position and moves up past the Earth, so the shadow starts at the top edge of the moon and moves down. If $V_m < V_e$ then the Moon starts at the top position and moves down, so the shadow starts at the bottom edge and moves up.
Note that my diagram shows the top view of the solar system i.e. looking down on the North Pole, so the top edge is the East edge and the bottom edge is the West edge.