I was wondering if it was possible to potentially grow a plant on a building taller than mount Everest. I mean on as in bottom to top with grass or some other plant. Would the grass run out of oxygen or would it live but be partially dead? If it is possible how could it be done?
Yes it could.
You have two issues here, pressure and temperature. For pressure, see this question on a sister site: What is the lowest pressure at which plants can survive? — it turns out that with the right kind of plant, this limit is at 1/10 of Earth 's sea level pressure; and with lichens you can even get lower, to the pressure of Mars. Since it's a building, I assume you want a garden. For small plants this is doable; you could have grass and some flowers.
The other issue is temperature. This NASA article suggests that by using bacteria genes we can have plants grow in really cold conditions. But hey! It is a building! You can shield the roof garden from wind and use chimneys to provide heat. With a building so tall you will have a lot of waste heat from water pumps, elevator engines, air conditioning and so on. Heating a garden seems like a really reasonable idea to put all that waste heat to good use.
If you want all surfaces covered with plants, that's also possible. The lower the better for them. Higher up you need to provide more heat. The Square cube law saves the day; with a building so big you're going to have more volume per surface square meter than modern buildings. This means heating the walls could make sense. Water would start to be an issue, but it is doable. A civilization able to build something this tall for sure would have pumps etc figured out. Make terraces every meter or so, with porous substance in place of earth and keep it moist for plants to use.
If you mean to grow on the outside of the building, like a climbing vine, I think the answer is probably yes.
Oxygen is one of the limiting factors, but the bigger one is heat, since a frozen plant can't grow. Both factors can be solved though. The upper floors will have to be heated and pressurized for human habitation. If the building isn't well sealed then warm, moist air will be leaking out, along with radiant heat coming from the building, which could be enough for the plant to survive, especially if it is something well adapted for high altitude. Over time roots would dig into the gaps, letting more air out, and giving the plant more warmth.
It would be horribly inefficient and expensive, but maybe they have really cheap energy and so don't care.
Not all the way to the top
Plants do not live at the top of the tallest mountains. Gymnostomum aeruginosum, a moss, was found at 6480 meters on Mount Everest; while Arenaria bryophylla is the highest flowering plant at 6180 meters on the same mountain.
Meanwhile, there are several mountains over 8000 meters. They are often covered in snow, so I did some investigation. I found a ledge of clearly flat, non-snow covered land near the top of Mount Everest at 27.98767 N, 86.91939 E. Checking with an altitude tool, this ledge is at 8220 plus or minus 20 meters, and about 100 meters across. There are many similar ledges nearby.
Since there are no mosses living at this altitude, or within 2000 meters of this altitude where there is evidently plenty of space not covered by snow in the summertime, I conclude that plants won't live this high as they are currently designed. If your building goes over ~6000 meters, there won't be much plant life that can survive on it.
Not a "conventional" plant.
A self-sustaining plant could not grow so tall because it would need to withstand too high a compressive stress (a column of wood 8 kilometers tall generates pressures in the order of the hundreds of atmospheres).
A vine, relying on the building for support, still could not do it because bringing water and mineral salts from the roots all the way to the top is a losing proposition. A 2m vine can need as little as half a liter of water per day, but a vine 8000 m tall would need somewhere around two cubic meters of water per day, which raises three problems: how to absorb that through the root system? How to supply the root system with enough energy to perform the task? And is there even that much water available to be pulled?
A lichen/moss carpet could. Nutrient transport could be as slow as needed, it would just limit growth rate; water would be gathered from the atmosphere, and sunlight is available everywhere. With no leaves, water loss and cold are much less of a problem.
Going up, dryness is going to be a problem, so we'd need some sort of slow water transport as well, and the lichen organization seems more promising: there are lichens surviving in the extreme cold. Low pressure and low oxygen are not a problem, they would just slow the growth rate, but the organism would simply gather what little CO2 might be available and accumulate it.
A plant needs to pump nutrients from the soil to every part of its "body", so you have a really big problem. You might be interested in Geoffrey West's new book Scale: The Universal Laws of Growth, Innovation, Sustainability and the Pace of Life in Organisms, Cities, Economies, and Companies, which goes into some of the issues. Also check out his TED talk. Tallness is an obvious advantage (monopolize sunlight), so if evolution hasn't grown a plant that big, it's a strong hint that it is very difficult - borderline impossible.
Me, I'd suggest a slight parabolic space holding water that has typical in-water plants growing at that height. The shallow water at the perimeter would warm the water to prevent freezing & providing movement would also contribute to that. Fish? The oceans are extremely deep & COLD & yet full of life all the way down. Would this body of water know where it was?