You'll want to heavily line your eyes in high quality liquid eye liner. As for the rest, you can buy an anti-sweating solution at most theatrical make-up supply places and then apply just about any kind of makeup over it. Personally though, I would recommend Ben Nye's MagiCake Aqua line of makeups. They breath, dry flat and smooth and un greasy, and should you perspire heavily, it still tends to more or less stay put.
If you're still worried about stuff getting into your eyes, you can lightly dab a little petroleum jelly (over the make up) around them to make liquids bead and roll away.
You can also get airbrush based makeup if you have an airbrush, and apply with that. It appears to be what was used in the photos you supplied.