Pretty much. Full disclosure: I have not been on NC yet, but yes, it's pretty simple I would go up to the spa reception desk, and say I wanted to use a locker and the facilities. The desk would give me a key for a locker, and ask for my room card, and off I would go to use the locker room and whatever else. When I was done, I'd hand back the key and they would give me my room card back. Easy peasy.
This is actually what I usually do on embarkation day. This does involve planning a little to bring your gym clothes in hand carry, but after lunch and whatever first day errands (adjusting specialty reservations, library, etc.), I usually get a locker, change, workout, and then steam/sauna/shower. By then, cabin and luggage are always ready!