Is the Radioactive Potato passed before the goal is checked?
No, they do not win.
The Looney Labs FAQs include a question about timing issues:

Q. Does the total effect of playing a card have to be resolved before checking for win conditions, or could someone win partway through?
A: As a matter of fact, yes, in Fluxx all consequences of a card play are considered to be functionally simultaneous, so there are a lot fewer timing issues than people sometimes think.
A good rule of thumb is that things in Fluxx (and many other Looney Labs games) are considered to happen simultaneously, so if there are timing issues, assume that things take place all at once, and then consider the outcome.

So the Goal change and the Potato move are simultaneous, meaning that the You Also Need a Baked Potato condition is not met, and the player does not win.