Germany

The castle is located in Fussen (142 miles away). It's about 6-7 hours by train.

Konstanz is probably the nearest German city to Zurich. Neuschwanstein castle is about 6+ hours from Zurich by bus or train. Fussen or Garmisch-Partenkirchen are probably the best German cities to go to to see the castle. Not a day trip option from Zurich.

Thanks y'll for sharing.

Just google Konstanz. it takes an hour to Konstanz from Zurich. See if I can stop by Appenzell on the way to Konstanz.

Are you talking about a day trip from Zurich? If so, I would say you could go to Apenzell or you could go to Konstanz. Research them both and see which you might like better. Going from Zurich to Konstanz and stopping in Apenzell on the way would take all day but it would be possible. Apenzell is not on the way to Konstanz. If this is a day trip from Zurich, returning to Zurich, then what is it you want to experience or see? Are you thinking about Konstanz just to say you've been to Germany? I guess I'm just a bit confused about what you are planning.

Sorry for the confusion. I will just go Konstanz directly from the airport and stay a night there. I am sure that I will not able to use the Swiss travel pass to Konstanz. But just double check.

If you want to see the outside of Neuschwanstein Castle, might be easier to travel to Anaheim, California after your trip and look at the replica at Disneyland.

Better use of your time would be seeing the Swiss Alps up close.

I am sure that I will not able to use the Swiss travel pass to Konstanz

Oh yes you can. Konstanz is a border town, the Swiss Pass is valid up to Konstanz, but no further. And not valid for travel in Konstanz.
See the official Swiss Pass validity map here: https://www.sbb.ch/content/dam/sbb/de/infotexte/uebersichtskarte-sts.pdf

Your orginal question confuses me.
"the nearest Germany city I can visit from Zurich" - why do you want to visit a German city? What are you looking for? Is this a day trip or do you want to stay there?
And what do you mean by a "city"? German does not distinguish between towns and cities, they are all the same word. It is personal opinion whether Konstanz is a town or city.
If you told us why you want to go to Germany and what you wanted to see we could suggest a suitable place to visit.

Neuschwanstein castle is at Füssen, but that is a long way from Zürich and not easy to reach from there. Too far for a day trip. And Füssen is definately not a city, it is a small town.

My money is on "I want a cheaper hotel room and I think I can spend less in Germany than Zurich - tell me where to go". Am I right?

I am trying to get a preliminary two week Switzerland itinerary put together. I just wondering if I can get a day or so in a Germany town. My flight will arrive at Zurich early morning and will not able to check in the hotel so I think I can go to Germany for a night (cheaper hotel) then back to my first leg of the itinerary of Lucerne ... still planning.

Minnie, thanks for the extra info.
So your plan is to go from Zürich airport to this "town in Germany" for one night and then on to Luzern, because you think that will be cheaper than an extra night in Luzern.
I am not sure that will save you much money. The prices do not suddenly drop at the German border. Places in Germany near the Swiss border have higher prices than the rest of Germany. And Konstanz is right on the border.
If you are arriving on a morning flight, Zürich airport to Konstanz is 1h 4 mins, you will still have the problem that it is too early to check into a hotel.
Zürich airport to Luzern is 1h 10 mins in the opposite direction. In Luzern station there is a left luggage office, which could solve your problem. Also, most hotels will store your luggage if you arrive early

To me, your plan is to "waste/throw away" a day to save a little bit of money. That is false economy. Pick somewhere you want to go (and Füssen/Neuschwanstein is too far) and spend your time there. This could be Luzern or somewhere else.
If you go to Konstanz, visit "Insel Mainau" a small lake island you can get to by local bus from Konstanz. Info: https://www.mainau.de/en/explore.html
If you are prepared to go further, Lindau is a town on the north side of the lake that is worth a visit. Reachable by train. Further from Zürich and Luzern. Info: https://www.lindau.de/en

I guess that not just the money saving but I will be able to visit a different country as well.