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Japanese railways operate on a fare zone basis: if you buy a ticket for X yen, you can go anywhere where the fare from your departure station is X yen.
For the specific case of the Skyliner, the Skyliner fare from Narita Airport to both Ueno and Nippori is exactly the same (2470 yen at time of writing), so the two are interchangeable. This means:
Yes. You do not hold a ticket "for Nippori", you hold one from Narita Airport for 2470 yen.
Yes. Transfers into other systems is a different kettle of dish though, a paper Skyliner ticket will not entitle you to enter the JR area unless you specifically bought a transfer ticket. If you use a Suica/Pasmo, though, the appropriate transfer deductions are computed automatically.
Yes, the Narita Airport stations are also both in the same fare zone.
Caveat: Limited expresses with mandatory reservations can complicate things slightly, because in addition to the plain ticket (乗車券) you also have a seat reservation (指定席) which is for a specific route. This may be why the staff asked you for your destination, although it's more likely they wanted to confirm that you don't intend to continue onward on JR. However, in the case of Nippori vs Ueno approximately nobody travels only that segment by Skyliner (I'm not sure it's even possible?), so the odds of somebody booking the same seat as you are nearly zero.