The option to upgrade varies across operators, but as a general position to upgrade you need to either call at a ticket office or ask the train guard. Some have an upgrade price but many will simply calculate the difference between a standard second class fare and the same first class fare and charge that. For the Zurich to Vienna service that difference is 102 euro per person.

I did that journey already, it’s very good in 2nd class. Difference between 1st and 2nd class on trains is usually not as big as on planes. Railjets are very comfortable.

To answer your question, I don’t seem to find a specific class upgrade ticket. Usually you pay the price difference between 2nd and 1st class with a member of staff, preferably before boarding (at the ticket office). 

Or you buy a full fare 1st class ticket for that journey, or you buy a full 1st class Eurail/Interrail. When are you travelling? With a bit of luck, you can still score a saver fare in 1st class.

If you want to know what to expect on the Railjet: https://www.seat61.com/trains-and-routes/railjet.htm#RailJet_journey_like_from_Zurich_to_Innsbruck,_through_the_Arlberg_Pass

You can also chose to change your Eurail pass to a 1st class pass if the conditions of your pass allow exchanging and your pass is not yet active - it is often not much more expensive than the upgrade cost for such long journeys.

Check your confirmation email to see if your pass is exchangable. If it is, this will incur a €15 exchange fee + the price difference.