Inventory:
Large piece of white sailcloth, nylon, waterproof*
Dark windbreaker, waterproof
Shirt
Shorts
Leatherman
Waterproof Matches

*By large, I think at least tarp size

Resources:
A cave
vines and vegitation, no trees
sea birds
marine life

A few thoughts...
The sailcloth is white, and reflects heat. The windbreaker is dark and absorbs it.

If you have to sew something for your project, you can probably make a bone needle from a dead bird. For cordage, you can make some from the vines and vegitation, or the sailcloth may have some heavy thread seams you can cannabilize.

The temperature inside cave can be significantly cooler than the ambient outside temperature. Perhaps this can be used to our advantage. It may even be low enough to condense the humid air and create dew you can soak up.

For small amounts of water, you can...
-Find out where birds hang out. There may be water in rock cracks and so forth.
-Collect dew in the morning with your tshirt
-Get some water from fish
-Probably get water from vines

For more significant amounts of water, the two obvious sources are the humid air, and the ocean though.