Yep, understand what you are trying to do, and I'll restate my original post.

First, try it for real. Pick up a cup and mimic the action you want to animate. The only reason the cup stays in position is because your hand pushes the cup against your lips, and when people move, they purse their lips to act as a cushion against the cup so they don't hit their teeth.

Now, break your scene down into it's stages. For one, I'd try parenting the cup to the specific finger (say middle 1) as that is where the true mass of the cup would be rotated from.

Next, make a pose where the cup is at his lips.

Finally, make a pose where the cup is at the lips but the head is turned. Aim for the cup to be about 6mm 1/4 inch away from his face at the end. Your last frame.

Next, go from origin to pose 1 as a keyframe about 2/3rds of the way through your total frame count. At 30 frames per second, about frame 60.

Then from around frame 75 or 80 enter the same keyframe/pose. That should hold him in position to take a sip.

Go to frame 110 or so, enter your last pose.

If the general motion is fluid, go back and tweak frames. Especially the last 30 or so, where the cup is against his lips. Track the cup, adjust his expression to purse the lips against the edge of the cup.

You should never need to change the constraints. Stay with the cup attached to the hand. His hand holds the cup, his face does not. Move the shoulder and collar joints to adjust position.