There are many parameters that go into answering this question. The short answer is: “Yes, mostly”. There are two things you need to know to get a more definitive answer: what is the angular profile of the fiber emission, and what does the manufacturer mean by NA? The emission profile is usually Gaussian, but not always. That is certainly the case for SMF. It is often approximately true for MMF, but it doesn’t have to be. Let’s say it’s Gaussian. The next question is what does NA mean? Often we expect a Gaussian to be defined at the 1/e^2 points. That would mean an aperture at the specified NA would pass 86% of the light. However, that isn’t usually how fibers are specified. Most manufacturers specify the NA as the cone angle that passes 95% (or sometimes 90%.). Suppose it’s 95%. Taking your NA and your focal length, 95% of the power is within ~ 4.5 mm. So your lens would be adequate to capture almost all the light.

Other factors are the angles that are launched into the fiber, the length of the fiber, and how coiled up the fiber is. The longer and bendier the fiber the less the launch conditions matter and the more you just get the specified cone. In most cases the launch conditions stop mattering after a meter or two.

Might I suggest just measuring the geometric part? Shine the bare fiber at a wall with sufficient distance that the cone is much larger than the size of some detector, but not so large that the detector can’t accurately detect the power, and then profile the beam. A camera without a lens is convenient if the wavelength works and the power isn’t too high. Once you know the beam profile you will be able to calculate what fraction enters your lens aperture.