Where did you bought your reservation? Eurail.com, ÖBB or another company?

Nightjet trains are often full, so maybe is it not possible to change. The Nightjet from Amsterdam and Utrecht ends half of the train in Basel and the other half continues to Zürich. 

Are you taking a seating car, couchette or sleeper carriage?

Another possibility is to continue the journey in the seating car (no reservation needed), or to ask staff on the train if you can go to Zürich (Reservation cost is the same for Basel or Zürich). 

It is NOT easier via Züri-only if you have the desire to stay in that bed you may have succeeded to get-as it is in fact a detour and on normal tickets you would then have to pay extra. Change in Basel SBB-train to Arth Goldau-and board that train from Z there. Nice ride btw.

PLus that the NJ is most often very late next morning on arrival-which may disturb all theoretical plans.

We made a reservation through Eurail, and for a couchette.

They advise me to book a new reservation to Zurich.. but obviously there is no availability. 

I think I will ask staff if we can stay/ change to seated car.

Thanks for your help!

Haha ok, so I'm letting go of my theoretical plans :)

tx!

There is no way to change dates and other details on your reservations after it has been paid for. The only way to do so is to cancel your current reservations in your account and to make new ones. Please note that the refund amount depends on the carrier.