Upper limit of neutron star mass and collapse to a black hole

I'll answer both of your questions in turn.

1. For your more general question, in classical general relativity, there is no lower mass limit to a black hole; you may make it as large or as small as you wish. For your more precise question, the upper bound to a non-rotating neutron star is the Tolman-Oppenheimer-Volkoff limit, which is between 2.1 to 2.3 solar masses. Beyond this, the neutron star will collapse into a black hole.

2. We do not yet have a perfect quantitative understanding of the interior of a neutron star, so currently this question is unanswerable. However, assuming our neutron star is a static, spherically symmetric mass made of a perfect fluid with a density that increases outwards, then we must have $$M<frac{4Rc^2}{9G}$$ where $R$ is the (areal) radius, $c$ is the speed of light, and $G$ is the gravitational constant. This is Buchdahl's theorem. So, if a neutron star obeyed the above (fairly reasonable) postulates, and was able to be brought right below the limit (which may or may not be the case), then it would be in the situation you describe; shoving in even a little bit more mass would inevitably cause collapse to a black hole.