The simplest way is to take a piece of rigid cardboard and a straight edge. Cut out the shape of your edge and drag it length wise down the edge.

In your case, apply a little more caulk than you need and you would cut out a 90⁰ section of the cardboard (round the corner a bit) and then start at one wall and work your way to the other. This should deliver the results you are looking for.

Edit: you should look for caulk removal tools as well

Patagonia will also repair items if you ship them to the factory, which is nice if you grow an emotional bond to things that go through things with you.

The metal would be fine if it's not corroded and you give it a proper cleaning. The platic might be brittle in places, if it was in the sun you should probably avoid using the plastic cup for hot things.