With a Neutron star, it will drop you in at only 0.22ls from the star. If your engines are up, you'll smack into the star before your realize, unless you are ready to dodge immediately. At zero throttle, you should be fine. Black Holes, however, will sometimes drop you at 4.05Mm! That mega-meters, not light seconds! Even at zero throttle (which is still 30km per second), you will still be approaching too fast, so swing out and get some distance before scanning the black holes.
When you land on a BH system at 0 throttle, you have about 3 minutes before getting too close. That's enough to blow the horn, scan the BH, give a look at system map, open the gal map and target next system of interest, realign and jump away. BH does not generate any heat, so worst case scenario you hit the body exclusion zone and get dropped from Supercruise. If your throttle was 0, you get 0 dmg and you can boost toward the BH up to 20ish Km. After that you can get closer by Flying tangential to it.