It would have to be near zero gravity, and you would have to have something even bigger to brace yourself against, and you would only be moving it an infinitesimally small amount at a time, but yes it would be possible.
That's ignoring a very long list of things that would kill you before you even got to try, anyway.
Weight is determined by gravity, which is determined by the mass of the objects.
Regardless of gravity, objects still retain their mass, so you wouldn't be able to move anything that massive.
Moving an object requires force, the amount of force required is related to the object's mass and current velocity (momentum). Even sitting still you'd have to accelerate the mass from zero.
I forget the acceleration formulas, physics was a few decades ago. F=M*A?
Yes, that's right. So the force required to accelerate an extremely massive object is very high.
But, if you only want to accelerate it a little tiny bit, you only need a little tiny bit of force. So all other things being equal, you could push on the sun and maybe after some days, weeks, months, or years, you'd start to notice that it moved a little bit.