Far longer than old age, unless you plan on living a very long time. The thing was only moving a few 10’s of kilometers per second.

It's possible that the first stage was also used to decelerate a bit before being jettisoned, explaining the relatively low speed

Not that low a speed. Let’s say the generational ship was moving 0.01c, at the low end of what’s believable (500-1000 years to a nearby star). That means the ship was decelerated 99.999% of the way from cruise speed before jettisoned. In terms of energy, that’s 99.99999999% of the stopping thrust already expended. That’s without even getting into the rocket equation which makes it even worse.

At that point why not carry it into the system with you? The mass would be negligible given the energy budget. Ejecting it seems to serve no purpose but wasting several cubic kilometers of presumably processed material. Why do that?

Now maybe it was a probe ejected from a ship at distance, but then why is it tumbling end over end?

> I figure that would probably have the real ship arriving probably sometime in my old age

Why would that be? Shouldn't the mother ship be not far behind this, depending on how much they've decelerated since detaching?

If the ship is decelerating, then the second stage moves slower than the first stage.

But even if this was true, the second stage would already be here.

As the first stage goes at tens of kilometers per second and as we did not detected the second stage it is probably at more than a AU, so it will need more than a million seconds to reach earth, only more than 11 days, actually.

> If the ship is decelerating, then the second stage moves slower than the first stage.

Only if you fire the next stage. Otherwise there will be only what little velocity difference was created by detachment itself.

> But even if this was true, the second stage would already be here.

Not if it was used to decelerate further :).