The surface area is measured in powder form (before making the actual anode) using the Brunauer-Emmett-Teller (BET) method. If you have the equipment, you can usually do a measurement in ~1 day. See the following for theory: https://en.wikipedia.org/wiki/BET_theory

Storage capacity is measured using a potentiostat. You can also do this with a voltage and current sources, but most potentiostats have software that is really helpful for battery analysis. The material also needs to be made into an electrode, and you need a counter electrode and electrolyte solution. In addition, most batteries need to be assembled in a glovebox.

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