sasha, Imagine a pump used to draw water from a well. It has a handle. To operate the pump you grab the handle and move it up and down. to pump....to cause to move with the up and down motion of a pump handle. to pump someone's arm ......to shake hands with someone in a manner that resemble the action of a pump handle. A hearty North American handshake has a brief up and down energetic HAND pumping motion. Pumping someone's arm is an exagerrated form of this hand shake. It is not necessary or even polite in the business world to pump someone's entire arm. Still, some people prefer the masculine energetic arm-pumping handshake, so expect to encounter it from time to time in America. Any native English speaker would understand that "to pump someone's arm" refers to "a very energetic, or sometimes overly aggressive, handshake".

To pump someone's arm or hand means to shake hands vigorously. It is a common phrase.