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If there was a satellite that had a periapsis at 250km, approximately how much delta-v would it take during each orbit to offset the atmospheric drag? Would it be different depending on apoapsis?
The idea is for a satellite to go through and pick up debris and then release it at periapsis
Putting it at 250km ensures that any debris will be removed and if the satellite brakes apart its debris will be removed as well.
The retrograde delta-v that the spacecraft will receive from atmospheric drag depends on two things: the mass of the spacecraft and the force of drag. (And the drag force will be integrated over the all of the time that the spacecraft is in the atmosphere.)
The drag force, in turn, depends on four things:
Spacecraft velocity will depend entirely on orbital characteristics, and air density will depend on altitude as well as time of day and climate/Sun conditions. And, in turn, the drag coefficient will add additional dependencies on the object's shape and the viscosity of the air.
Would it be different depending on apoapsis?
As you go through the atmosphere, many of the properties above will change, and based on the orbit, your velocity at different parts of the orbit, as well as what parts of the atmosphere you're passing through, will change as well. With a fixed perigee, the position of your apogee will determine the overall shape of the orbit, which will change all of these factors. So yes, the acceleration due to drag that the spacecraft experiences will vary drastically depending on the orbit.
In short, you'll need to specify more characteristics of the orbit in order for your question to be answered properly. However, it's possible to approximate an answer to the question if you assume a circular orbit (i.e., apogee distance = perigee distance) and make lots of assumptions about the shape of your craft.
If we go by the tables at the end of Space Mission Analysis and Design, if you have a spacecraft that has 50kg of mass per square meter of cross-sectional area, and you're in a 250km circular orbit, then you must impart a delta-v of between 636 m/s and 2,002 m/s each year (depending on solar activity) in order to maintain altitude; i.e., you can lose that much delta-v during the course of one year. If your spacecraft is more massive, then it will lose velocity more slowly (as it has greater momentum).
I agree with the other comments, however, that your approach seems to be somewhat flawed, as you'll need to impart a delta-v to the debris lest it simply continue to follow the same orbit you're in. (It will eventually deorbit on its own, sure, but very slowly.)
