In Germany, Vienna and Switzerland you can use the S-Bahn with your pass. You need to add the trains to your trip in the normal way. Obviously it’s not good use of a pass day unless you are doing other travel that day.
But in Berlin, you are likely to want to use the trams and U-Bahn. They aren’t covered by the pass, so you might end up buying a day pass anyway.
There are two S-Bahn line not covered, and they’re in Zurich.
I don’t know about Prague or Budapest.
In Vienna, the S-Bahn passes to the East of the city, and misses out on the tourist area. So again, you will probably want to buy tickets or day passes so that you can use the U-bahn.
The same is true in Frankfurt.
In general Eurail is somewhere between poor value and completely useless for any day you spend within the confines of most cities. It is only valid on railways, not underground, metro, tram, bus or boat transportation and the extent that the rail network is useful for city travel varies massively. S-Bahn is the name given to these commuter rail lines in Germanic countries
Eurail is really only of value for reasonably long journeys in most countries, I would say you should expect only to use it for getting between these cities and look for local tickets for the cities themselves.
In fact, if you have already bought night tickes for Berlin - Budapest as standalone tickets it may well be cheaper for you to buy regular tickets for the other 3 city-city trips rather than a Eurail pass.
Most cities have day tickets of some sort for their local transport systems.
That “Budapest Card” at €30+ is a tourist pass that combines a day transport ticket with entry to various attractions, you would have to calculate if it good value for your specific plans in the city.
The cost for a transport 24hour ticket is 2500HUF, approx €6.60, there is also a 72hour ticket for 5500HUF, €14.60
https://bkk.hu/en/tickets-and-passes/prices/
In general it is always a good idea to look at the tickets offered for normal travel as well as products marketed at tourists, it will depend on how much value you get out of the added cost of the tourist attractions to make the tourist cards worthwhile.
Thanks for your responses. You have generally confirmed what my research had shown. I’m going to post a separate question regarding any day trips from Prague, Berlin, Budapest, and Vienna that are worthwhile in December. Once I have those responses I can decide what Eurail ticket level to purchase.