What is the proper way to wash after a Funeral or a visit to the Cemetery?
The Shulchan Aruch O.C. 4:18 brings a mixed list of things that require "washing". Some require water for Ruach Raa, others for cleanliness preceding davening, etc. The S.A. there brings an opinion to wash after walking among the dead (a cemetery).
The Mishna Berura there (39) says that the only area that absolutely requires 3x/hand is when one wakes up, but that some are strict by cemeteries (and tashmish and perhaps bathrooms).
I don't know of any source that requires a cup except by morning tefila and bread.
So while everyone is waiting for the single cup, I walk to the faucet and alternate my hands 3x each under the water.
After attending a funeral, one should wash both hands three times in the above-described manner (Machatzis HaShekel 4:17). The custom recorded by early poskim is that one may not enter a building after touching or escorting a meis without first washing netilas yadayim (Rama, Yoreh Deah 376:5). After this ablution, the custom is to turn the cup upside down and put it down rather than hand it to another person (Eliyahu Rabbah 224:7; Chochmas Odom 158:30; Rabbi Akiva Eiger, Comments to Yoreh Deah 376.
All those who attended the funeral must wash their hands ritually, once outside the cemetery area. Take a large cup of water in the left hand, pour it over the entire right hand, covering up to the wrist. Take the cup in the right hand, and pour it over the entire left hand, covering up to the wrist. Repeat two additional times. It is customary to place the cup upside down after washing, and not to dry one's hands with a towel or paper, so that the memory of the deceased lingers.