How to physically clean a USB connector safely

Isopropyl alcohol (aka: Rubbing alcohol). You may be able to buy some at a local pharmacy/chemist, but I usually have to get mine online. It's great for cleaning off cooked-on thermal paste - it shouldn't have an issue with some sticky finger stuff.

Use some compressed air? Or some Cyber Clean?

Try using alcohol swabs. I have found them useful for cleaning keyboards, sd memory cards, data sticks, and they work great on cleaning laptop screens as well. After cleaning the laptop screen, buff the the screen with something non-abrasive (a kleenex or a cloth diaper).