If you have fresh pomegranate juice, keep it refrigerated and use it within about 14 days from the time it was pressed. If you're buying fresh juice from the store in the refrigerated section, use it within 14 days from the time you open it.
If you think you won't use up your juice and you aren't ready for a martini, freezing it is the best storage option. Freezing is the best way to preserve the flavor of shelf-stable juice too, once it's been open for two weeks. The properties that make pomegranate juice so good for you can degrade over time as well. That means you'll lose nutrition and flavor.
Orange Flavored Liqueur
Orange or Tangerine Slices
You'll also need a slice of orange, tangerine or mandarin. I say "optional" in the recipe, but the fresh citrus makes a big difference, especially in the aroma of the drink. A twist or citrus zest will also work.