Firstly, my apologies for my earlier comment, I did not read the title of your question correctly. Please let me know if if I have misunderstood you again.
To lift something in the Earth's atmosphere, we need to remove the air from the aerogel, (as small as a volume of air as it is), to create a vacuum, otherwise we still have a heavier than air structure. Then we need to seal the sides of the aerogel with graphene, to maintain the vacuum. At least, I hope I understand your idea properly here.
This source : Graphene Properties
Graphene aerogel is also very light at 0.77 milligrams per square metre (for comparison purposes, 1 square metre of paper is roughly 1000 times heavier). It is often said that a single sheet of graphene (being only 1 atom thick), sufficient in size enough to cover a whole football field, would weigh under 1 single gram.
This is incredibly light, and the slightest breeze would mean trouble in working with it, but would it be buoyant in air?
The problem seems to me to be mainly structural integrity. Graphene may be strong in certain directions, but I am sure you have seen what happens to a steel can, when you create a vacuum inside off of it. You can take the aerogel and remove as much air as possible, and seal the sides with graphene, but this only increases the chances of crushing by normal air pressure as you create a "harder" vacuum. The displacement by volume, creating a bouyant force, will also unfortunately collapse.