Most times, strops can just be wiped down with a dry paper towel. Don't worry too much about steel residue making the surface black or 'dirty looking'. The strop will still do it's job. I would recommend getting some extra compound, if you don't already have it. If there comes a day when you do feel the need to really clean it up, a paper towel moistened with WD-40 usually works pretty well (don't soak the leather in WD-40, just moisten a paper towel and wipe the leather with that). If you have additional compound on hand, you won't have to worry anymore about scrubbing the old stuff off. That's part of the normal 'life cycle' of a working strop anyways; use it, clean it, and occasionally strip old compound off and renew it. Shouldn't have to strip it very often though. Most 'occasional use' strops can go a very long time, before needing any major cleaning up.
For storage, there's no need to get really fastidious about it. Just wipe it down with the dry paper towel, and store it inside a plastic bag or something similar. Doesn't have to be airtight; no harm will come to it, if it's allowed to breathe a little. The main goal is just to keep other dust/dirt/debris from settling on it. When sharpening your knives, make sure to wipe the blade down after the stones, so none of the coarser grit from them gets transferred to your strop. And if using your strop on a bench, make sure the work surface is clean, so the 'down' side of the strop bat won't pick up dirt/grit from the benchtop.