The grounds are open all year and free. You can walk to the fence from the Nauset Light Beach car park in a couple of minutes without paying a beach fee if you are not using the beach, though parking during the summer season is another matter entirely and the lot closes when full.
Climbing is a different question. Interior access is run by the Nauset Light Preservation Society on a volunteer schedule rather than by the park, and in recent seasons that has meant Sunday afternoons from May to late October, with additional midweek openings in July and August. Those days change, they are weather-dependent, and they are not something to build a trip around without checking the society's own schedule first.
The climb, when it runs, is a steep interior ladder-stair rather than a staircase with landings, and the lantern room is small. Expect a queue on an August Sunday and expect a minimum age or height rule to apply.
The gate to the grounds is open outside tour times, so an out-of-season visit still works. February at Nauset Light, with the wind off the Atlantic and nobody in the lot, is arguably better than August.