The intent of the original design was to use the shower head as the sink faucet as well. There should be a place to hang the shower head such that it directs its flow into the sink for that purpose.
What's important for installing an aftermarket faucet is the spacing between the two stems. AFAIK most of them are the same. Simply measure the distance from the center of one faucet handle to the other, or better yet, from one inlet to the other underneath, and compare that to the specifications of the new faucet you are looking at. Generally the inlets and handles line up with each other, and that faucet looks like the same generic faucet on my Palomino's shower.
Once you've found your new faucet, it is just like changing the faucet on your bathroom sink at home. Turn off the water, disconnect the lines, remove the nuts from underneath that hold the faucet to the sink, pull the old faucet, and reverse the steps to install the new one.