Can 200 watts lift a 100 kg box, and if so how long would it take?

The only definite number we can tell you is that in order to lift
a 100kg box, you have to somehow provide a force of about
980 newtons (about 220 pounds) pointing upward.  That's the
weight of the box, and you have to lift with at least that much force
just to get the box off the ground.

After that, the answer to your question is 'YES'.  The time it takes
depends on how much force you have available, how far you want
to lift the box, and how fast.

Here are the Physics formulas you have to use:

-- The amount of "Work" you'll do is

    (the amount of force you use) times (how far up you lift the box) .

-- The "Power" (amount of watts) you'll use is

    (the amount of 'work' you do) divided by (the length of time it takes).  

So ...

-- The higher you lift the box, the more 'work' your lift-machine does.

-- The faster the box is lifted, the more 'power' your lift-machine needs.