Eating and drinking in a cemetery:
It is forbidden to eat or drink inside of a cemetery. [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 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 Tzadiks 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 Teruma 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 Tzadik
 So is the widespread custom in Eretz Hakodesh near all the Kivrei Tzadikim that have their own Tziyon.