The chemistry laws cited by Robert apply, I believe, to gases in equilibrium. The key to achieving a CO2 blanket is to provide non-equilibrium conditions. Anyone who has worked with cryogenic liquids knows that oxygen displacement is a real thing and potentially very hazardous. You can be dead long before equilibrium is reached.
The keys are the flow rate and the density difference. If you blast CO2 from a pressurized cylinder into your head space it will generate a lot of turbulence and the CO2 and O2 will mix and it will not generate a blanket. If you flow the CO2 gently in to the bottom of the head space it will pool without mixing and gradually push out the air. You need to do this fast enough to avoid any diffusion or approach to equilibrium but slow enough to avoid turbulence. it is hard to judge this with an invisible gas.
Cooling the CO2 increases its density and enhances the effect. Cooling it enough that it condenses water vapor into fog makes it visible so you can more easily see what is happening. If you drop chunks of dry ice into your carboy you will generate very cold CO2 at the bottom that will push out all the air and make fog that rises out the top and then sinks down around the edges and gradually dissipates. This allows you to see where the CO2 is. Look at this video: https://www.youtube.com/watch?v=6kxc5yXN80Q . How much O2 do you think is left in that bucket? Not much at all.
So, to answer your original question, the CO2 pressure should be very low (to avoid turbulence), and there is no real way to estimate the amount of time required. If you had a flow meter you might have a chance of knowing by transferring enough CO2 to fill the head space 3 times over, or something like that.
To add to Richard's explanation. you also need to look at Boyle's Law of partial pressures. When you introduce a second gas into a container containing a first gas (i.e. introducing CO2, second gas, into an atmosphere with Nitrogen, Oxygen, CO2, etc. first gas) the gases will mix according to the partial pressures law. No matter how much CO2 you push into the container, there will still be some residual O2 that will mix with the CO2. The only way to get everything out of the carboy is in a total vacuum. Yes, since CO2 is heavier than air you will get most of the O2 out at a low introduction pressure into the carboy but some mixing will still occur, it is unavoidable.
Denny is correct that the easiest way to get most of the oxygen out of a carboy is to fill it with sanitizer and push it out with CO2 (REMEMBER, no more than 2.5 PSI so the carboy does not explode and cause injury... see one of my earlier posts on fermentation vessels). There is a caveat however. The sanitizer will also have dissolved O2 in it so you will not get a complete purge, but you will reduce the level of O2 in the container significantly to the point that the oxidation of the beer will be greatly minimized. Most likely, you will consume all of the beer before the oxidation effects of any remaining O2 will become apparent. I always say, do the best you can and RDWHAHB.
A lot of breweries that bottle their beer use a double pre-evacuation of the bottles prior to filling them. That is, the shoot CO2 into the bottle then suck it out to create a vacuum in the bottle. They then repeat this process before filling them with beer. However, doing something like this on a homebrewer's scale with a carboy is obviously not practical from an expense and danger of implosion standpoint.