NETA ATS Section 7.19.2 will give you the best guidance on this. I can summarize to give you an idea but be sure to refer to the latest standard.
Visual/Mechanical: Verify nameplate matches project specifications, inspect the installation and appropriate clearances, verify individual ground leads on each device, check stroke counter if present.
Electrical: Measure resistance through bolted electrical connections and grounding connections using low-resistance ohmmeter and perform insulation resistance test. Watt-loss testing is optional per NETA but I would recommended if available. Test voltages for electrical tests will depend on the voltage class of the arrester.
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