Cleaning and wipe down?

i try not to oil, clean, or work on my guns with the stock on ... that said, the procedure thats been working out for me a long time is to use air venturi metophilic oil to treat JUST the metal on guns occasionally and when new from corrosion avoiding any plastic or rubber and then when dry as part of regular maintenance keeping them wiped down with silicone oil which is easy to keep wiped off and even on everything .. 'any' other types of oils ive found to cause some kind of issues on air guns ..
 
It sounds like you are asking about wiping down the exterior of the rifle and stock, not internal cleaning or lubrication. If so, it doesn't much mater what you use. Assuming some of the steel surfaces are subject to rust, just use something appropriate, Ballistol ,FP-10, Breakfree, etc. Regarding hand prints on a synthetic stock, try lightly dampening your towel. I can't say for sure, I've never been much concerned with the surface of a synthetic stock, I figure that's why it's synthetic!
 
I buy cheap micro-fiber cloths for lots of purposes. I also buy spray silicone lubricant for lots of purposes.

For wipe down of my AG and PBs, I simply spray one of those new clean cloths with (CRC Heavy Duty Silicone Spray from Walmart is what I use most of the time, but even WD40 works fine) the lubricant and saturate it. Allow time for the volatile aerosol accelerants to evaporate and place that cloth in a plastic zip lock bag. Then I use that for the final wipe down on all of my CLEAN guns and ONLY use it for that purpose replacing it in the plastic zip lock bag when not being used. I replenish it once or twice a year with silicone spray.

This has worked very well for me for over 40 years. YMMV

Kerry