When you paste the AI stroke into a layer in AE(new solid or other) it will become a mask. You’ll apply the Stroke to that mask.
If you have a hang up pasting into AE, Then in AI: Edit>Preferences > Files and Cipboard, and make sure the AICB radio button is checked (mine has PDF checked (AI 10) and it also works).
You can edit masks in AE by hitting “m” on the keyboard, to reveal mask shape.
Selecting a vertex (the little nodes/points), will bring up handles to change it the curve, and you can drag the vertex. Double clicking a vertex will give you a transform box like in Photoshop.
If you want the mask shape to change over time, keyframe the changes over time.
The Pen Tool in AE is very similar to AI’s pen tool for adding and removeing points. If you’re familiar with AI, you’ll get it right away.