This question touches on the distinction between weight and mass which are used confusingly synonymously in many Western everyday-languages (not so in e.g. Russian).
Weight is the force of gravity pulling you down. It is proportional to the object's mass which explains the confusing everyday usage of the word. A bathroom scale really measures the force ("weight") you push down on with, but it displays a mass in kg or lbs. It does the conversion assuming Earth surface gravity which is where most (but not all) bathrooms are located.
In weightlessness (note there's a subtle difference between "zero g" and "weightlessness"), your bathroom scale wouldn't work because there is no force ("weight") to measure, hence the name "weightlessness". You'd still have the same amount of mass. And it is this mass which doubles as a source of inertia. That is, a massive (not heavy!) rock in outer space would still be "hard" to push in that you have to invest a lot of energy to get it moving. However, since there is no friction or air resistance (assuming vacuum), there are no losses you have to overcome constantly. A small but steady force applied over a long period of time will eventually get that rock moving ever more faster.
So even if superman could make an object weightless on contact, he would still need some muscle to get it to move. But not having to keep up with gravity, he can move it off the ground, thereby bypassing friction with the ground, so the object can actually get ever more faster without loosing the velocity immediately again. (However, air resistance would still be a thing.)