Eating and drinking in a cemetery:
It is forbidden to eat or drink inside of a cemetery. [This applies throughout the entire cemetery, even if one is a distance of four Amos from the graves, as explained in the Q&A. This applies even if one already said a blessing elsewhere and will not have to recite a new blessing in the cemetery. All the more so is it forbidden to say a blessing over food in a cemetery.]
May one enter food into a cemetery?
There is no prohibition against entering food or drink into a cemetery, and the food does not contract impurity or an evil spirit through doing so. It goes without saying that there is no issue with eating foods and drinks that entered a cemetery.
May one eat or drink in the cemetery if one is a distance of four Amos from any of the graves?
May one eat or drink near Kivrei Tzaddikim?
If the grave is inside a cemetery, then it is forbidden to do so due to the other graves that are around. If, however, the Tzaddik’s burial place is in its own grounds, and not within a cemetery, then seemingly it is permitted to eat and drink near the Kever, or inside the Tziyon, and so is the widespread custom.
 Rama Y.D. 368:1; Semag
 The reason: Some Poskim rule it is forbidden to eat and drink in a cemetery due to it being belittling of the dead. [Biur Hagr”a ibid; implication of Rama and Michaber ibid; Implication of Megillah 29; Aruch Hashulchan 368:1] Other Poskim rule it is forbidden being that cemeteries are forbidden in benefit, and hence one may not use it to benefit from food. [Gilyon Maharsha ibid; Rambam 14:13]
 Poskim in Nitei Gavriel Aveilus 92:4 footnote 5; See Eiruvin 31a “One may make an Eiruv for a Kohen with Tahor Terumah in a cemetery” thus proving there is no issue of impurity entering the food.
 Admur 45:1 regarding Tefillin and the same would apply in this regard; Implication of wording of Michaber ibid that the prohibition applies to the entire cemetery; Chasam Sofer 335, brought in Pischeiy Teshuvah 368:1 regarding areas that have been designated as a cemetery, and certainly this would apply to all areas of the cemetery that are designated for burial.
 As Tzadikim are considered alive even when dead, as we find a similar allowance to Daven and wear Tefillin near the Kever of a Tzaddik
 So is the widespread custom in Eretz Hakodesh near all the Kivrei Tzadikim that have their own Tziyon.