Imagine a huge fuel depot in geostationary orbit. Suppose further that it is tethered to the Earth and that there's a large hose snaking down all the way. Is it feasible to pump fuel up 35,786km? Can something based on capillary action work?
I can't imagine such a system would make any sense to build.
At it's heart you have a space elevator--something we don't have the materials to build on Earth yet.
A pipe full of fuel hanging on that elevator will be HEAVY--meaning a lot more elevator. Not to mention the weight of all the fuel in it. The pressure requirements won't be a big deal--use booster pumps like other pipelines do.
The thing is there's little reason for it--if you have an elevator why not just haul fuel tanks up it?
Capillary action will not get your fuel to space as this article in Wikipedia points out the height of lift is related to the diameter of the tube, where a narrower tube gets you more lift measured in inches. Trees use capillary action combined with other actions to raise fluids. In no case would capillary action by itself get you anywhere near orbit.
Assuming you have a space elevator and a line running the length of it, you might think you could "pump" your fuel for "free". As seen in Benefit of sling shot effect with a space elevator once you get to twice geosynchronous radius, you have more outward then downward pressure. In theory you could siphon off the oceans of the world without any cost above establishing the line and priming it.
Alas a siphon will not work, the top fluid is pulling up the lower fluid, so you would need to exert the same amount of pull that would be required to push or pump the liquid up your elevation. There are many variable; atmospheric pressure, fluid density, purity of the fluid and several other factors.
At distances measured in thousands of kilometer's, regardless of the other variables you would encounter, at some point you reach the Vapor Pressure of the fluid, and it would turn from a liquid into a gas. At this point the siphon stops functioning, Wikipedia has some formulas for what point this will occur.
As for directly pumping, there are a lot of variables, density of your fuel, diameter of your line, turbulence, resistance and such, but in the end a column of fluid 35,786km high would be VERY Heavy, it is difficult (impossible?) to imagine a single pump performing the task.
You could of course have pumping stations, with several pumps at whatever level your engineering solution desires, but this leaves you with the second huge problem with a space elevator which is getting power (presumably electrical) along a 35,786km stretch.
If your goal is fuel at GSO, go with Loren Pechtel's solution and haul tanks of fuel in the elevator.