There is no "minimum" for NASS. You take the CFA there but, as indicated, you can take it again and submit the new results.
I will say that USNA looks carefully at pullups because they believe that upper body strength is a strong measure of success in the physical aptitude portion of the curriculum. As a male, you want to be able to do at least 5 pullups. More is better; fewer is not necessarily fatal. For a female, one pullup is better than forever on the flexed arm hang and considered good. Obviously, the more you do the better.
There are various programs/recommendations, etc. on improving upper body strength in general and pullups in particular. I'm sure most of them work, but they aren't designed with your current level of fitness and goals in mind.
Thus, IF you have the option, I would contact a coach, gym teacher, or personal trainer to discuss your specific goals and have that person put together a workout program designed specifically for you. It should also include running, unless you already excel at that. As you know, major improvement won't happen overnight but it will happen.
If you don't have access to an individualized program, something along the lines of what is discussed above will certainly do. The key is to stick with it. It's not just about doing well on the CFA, it's about getting into shape and staying in shape for plebe summer and the rest of your life.