Is there a simple way an amateur like myself can do some kind of a cleaning on a movement without taking the watch apart bit by bit?

NO! Sorry, but there are no acceptable shortcuts IMO.

Afraid of that, just do not have fine motor skills it takes to do it.

I will strongly agree with Roger. You never know what defects you will find in an old watch that must be attended to. Even if you had a 100% running watch in perfect condition, a "shortcut" cleaning method would actually cause problems.
Mike

quote:

Afraid of that, just do not have fine motor skills it takes to do it.

you don't need fine motor skills. Just a fat wallet and a trustworthy watchmaker