Depends a little on how exactly you assume this happens.

The sun won't go supernova, it doesn't have the mass, so regardless, we're talking about something that's theoretical.

If you assume "All the mass of the sun is suddenly *boop* in a singularity in the current center-of-mass of the sun" then Earth, the planet, would be fine. We would last a few days or weeks before the Earth cooled off enough to become uninhabitable. People might be able to survive for quite a while if they were in insulated, completely-sealed, perfect-recycling buildings with a long-lasting power source.

If you assume a slightly more realistic scenario like "a very small black hole wanders in from outside the solar system, hits the sun perfectly, somehow stops while inside the sun, and starts eating it from the inside out" -- I believe the accretion disk radiation would fry us pretty quick, but I'm not sure how to calculate that.

(Which brings up a point -- if a very small black hole did fall directly towards the sun from interstellar space, would it actually stop? Concepts like friction don't really apply to black holes. I guess the singularity would get more inflowing momentum from its direction of travel, so it would slow down somewhat...)