I think coldmanbrew might need a longer answer because I believe a cup of honey is about twice as much as you'd need for a 5 gallon batch. Honey is best measured by weight if you have a scale.
Carbonation is best calculated with a carbonation calculator because it is variable depending on temp. Assuming a typical carbonation in a cider bottled at room temperature, you'd probably use about 5 oz (by weight) of sugar, which is about 3/4 cup. Honey is about 80% sugar, so 5 oz of sugar is equal to about 6 oz of honey (by weight). To convert to volume, multiply by 2/3, which gives you 4 oz (by volume) or a half cup. The reason for the volume correction is that a gallon of water is about 8 lbs and a gallon of honey is much heavier at about 12 lbs. 12 lbs is 192 oz (weight) and a gallon is 128 oz (volume). 192 x 2/3 = 128. I hope this makes sense. Bottling at a cooler temp would require even less honey.
Keep in mind that carbonation will take a while longer than normal, as honey takes longer to break down and be consumed. Also, if your cider is high abv, the yeast may be stressed out, which will take even longer to fully carb.