I have basically that same fixture on my boat, and it is self-priming. It has to be, since the water tanks are generally mounted far lower than it. Most of the manual pump faucet fixtures on boats use a self-priming pump of some sort.
If there is water in the cooler sump, then the valves are not in the correct positions. You may want to try changing the positions of the valves, after you confirm that the faucet does indeed work. If the faucet isn't plumbed into the boat's fresh water system, disconnect the hose at the bottom and put a temporary hose on it into a dish or bucket of water... and try pumping the handle. If water comes out, then the faucet pump mechanism isn't the problem.
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You know what the first rule of sailing is? ...Love. You can learn all the math in the 'verse, but you take
a boat to the sea you don't love, she'll shake you off just as sure as the turning of the worlds. Love keeps
her going when she oughta fall down, tells you she's hurting 'fore she keens. Makes her a home.
—Cpt. Mal Reynolds, Serenity (edited)
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Still—DON'T READ THAT POST AGAIN.