How did a Soviet SA-3 missile manage to shoot down the F-117 Nighthawk?
"Stealth" does not mean invisible. In fact, there are photographs of F-117s, so clearly, they are visible.

Stealth means less observable than a comparable non-stealth design. A stealth aircraft still has a radar signature, it still generates heat, it still generates noise, and it is still visible … just less so than a non-stealth aircraft.

In this particular case, there was a combination of several factors:


The F-117 was the first production stealth aircraft. A lot of things weren't known by then, for example how to integrate sensor pods, radomes, etc. into the faceted hull in such a way as to not negate the stealth characteristics. As a result, the F-117 was pretty starved for sensor data, for example, it didn't have a radar warning system. The pilot literally didn't even know that he was overflying a missile station until he saw the missiles coming at him through the clouds.
The F-117s were flying exactly the same route at more or less exactly the same time night after night. That's just shoddy mission planning.
The bomb bay is a huge radar reflector, with the doors open, the plane is no longer stealthy. (This is still true for the B-2, BTW.)


So, since they flew the same route every night, it was likely that someone, somewhere would just simply spot or hear them by accident. No matter how stealthy your aircraft is, if someone looks up and you pass between them and, say, the moon, they will see you. Since the route was known, the Yugoslav Army could move their missile station right into the flight path. Since they were also repeatedly bombing the same targets, the Yugoslavs could even make an educated guess at which point the bomb bay doors would be open. They also only had to turn on their radars for short periods of time (again, since they already knew roughly when and where the airplane would be), making it harder to detect them.

And once the missile hits you, it doesn't really matter how old it is. Kaboom is kaboom, regardless of age.

So, to summarize, it was a combination of


bad mission planning
limited sensor capabilities of the F-117
a very clever missile commander (credit where credit is due!)