The train is fully booked on the 5th of June. There are couchettes and sleepers available on the 6th of June.

Is there any alternative for me to travel from Amsterdam to Zurich on 5th June night?

Yes you can take the same night train but only as far as Basel. There are couchettes and sleepers available. From there stay on the train but move to seating carriages to Zurich (shouldn't count as using a second pass day as you're staying on the train).

To book night trains use tickets.oebb.at or nightjet.com (no booking fee).

It's always best to avoid eurail.com for seats because they are often more expensive and charge a 2€ booking fee.

Reservations are optional in Switzerland, Germany, Austria and some other countries. A complete waste of money in Switzerland but recommended on busy days in other countries.

General advice for seat reservations :

https://www.seat61.com/interrail-and-eurail-reservations.htm

https://community.eurail.com/train-connections-reservations-47/how-to-get-reservations-105

You can use the seating coaches in IC60403 departing Amsterdam Centraal at 20.28 and arriving in Zürich at 8.05 on the 6th of June. Book at DB (4,50 EUR) or maybe at ÖBB (3 EUR).

Thanks all, @AnnaB ,  I have reserved as your suggestion.