Unexpectedly heading to Zurich for a few days - Help!

Given the difficulty I've had finding even moderately priced hotels in Switzerland, I have to doubt that there are any cheap hotels in Switzerland. If you have the time, consider a daytrip to Luzern/Lucerne. It is an hour from Zurich and spectacular. You could do a boat ride and a cog wheel train ride up to Mount Rigi, or just wander Luzern's gorgeous old town area.

How many nights? What are your interests?
Download the SBB app for trains. It's an easy and quick train from the airport to Zurich old town.
(The S-bahn is Germany.)
Depending on the length of visit, the half -fare card may be best for trains. Most hotels give you a "mobility card" for free local transportation.
I'm not a fan of Swiss food, and everything is expensive! I'd recommend an hotel with "free" breakfast, eating big late lunches, and grocery store dinners if you're on a budget. Try the rooftop Coop store cafe for lunch.
If your daughter is going to college there, prepare.
I second Jules' recommendation of going to Lucerne, it's a charming city.
Please supply a few more details and I may be more helpful. Enjoy!

Zurich is one of the most expensive cities in Europe, if not the world. Nothing is cheap there. I suggest trying ALL the sites: Google Maps, Booking.com, AirBnB. Ask the folks at the school what they recommend. Don't rule out hostels, either. Maybe try going OUT on the tram line instead of closer to the center.
As far as activities, I would say that if your daughter is serious about attending this college, you should spend your time there finding out as much as you can about what her experience there would be like, especially the availability and cost of housing. If she attends, you'll have plenty of time for tourism while visiting her, right?

You can simply take the train from Zurich’s Airport into the city center. From there, you’ll need to transfer to local trams to reach the college area. Staying in a hotel room with a shared bathroom will cut accommodation costs considerably. In restaurants the service charge is included in the bill— so no need to tip. Grocery stores are the budget strategy, along with pizza in restaurants.
Everything in Switzerland is expensive— housing, groceries transportation and utility bills. Zurich is tied with Singapore as being the most expensive city in the world. This visit will be an eye-opening. (and possibly eye-watering) experience. Get ready for being shocked at the prices in Zurich—- they may have you or your daughter rethinking her college plans.

If you’ll be there over a weekend, I would actually avoid Langstrasse as it’s one of the main nightlife areas in Zurich. Can even get a bit seedy in some spots. Anyway Zurich is not that big, so you can stay almost anywhere in the city and easily get over to Toni Areal, which itself also has some hotels that are ok.
Re hotels, cheap and Zurich are not words normally used in the same sentence. Although if you are coming soon, should be a bit better than in summer and you might get a relative bargain. Look at one of the Fred hotels, Citizen M, Motel One, or Adagio. Josephine’s Guest House for women is also a good value for female travelers. You could also even check the Mercure at the end of the #4 tram line (which stops at Toni Areal) near Altstetten train station. Easy to get into Zurich city center or airport.
I would only take the tram from the airport if your hotel is at a stop along the way.
Activity suggestions would depend on how much time you have. If a short trip, you probably need to spend most of your time figuring out Zurich and the school itself, where your daughter would realistically live, see how much rents, groceries etc cost. If you have extra time, go to Luzern for the day, one hour each way. Boat ride in Zurich is a an option if on a tighter schedule.

congratulations to your daughter and to you

sure, it may be (it is) that Zürich isn't especially inexpensive, but it is a very pleasant city to be and live in. Nice scenery and very pleasant lake. Clean and neat generally. Good recreation areas, even an occasional steam train, and a great transportation hub.

My experiences are from personal visits, a good friend who teaches at the equivalent of entry university level in Zürich and who lives a short train ride away, and another friend who lived and worked for a bank in Zürich several years ago.

If you take it for what it is, and your daughter must be doing that and the uni meets her needs, it can work out very well.

Does she speak high German? High Swiss German is a little different, and the dialects quite different, but Zürich is a very cosmopolitan city. The business district is quite small, and has lovely little park squares throughout.

Appreciate all of the advice!! We booked a small room in a little hotel in the old town (recommended by Frommers) for $150/night. It's near a #4 tram stop so I'm hoping it will be easy to get to the Toni area.

Good advice about pizza and grocery stores.

We'll only have 1 or maybe 2 full days to wander around. Going in mid-April. It looks reasonably warm there now, but will probably be chilly and rainy. I'm thinking we'll walk around the old town, check out some churches, maybe the Landesmuseum, maybe a boat ride, weather permitting?

IF (big IF) she gets into this school, the tuition is so much cheaper than in the U.S., it will balance out with the expensive housing costs and food.

Thanks again!

Thanks for the advice to avoid Langstrasse on the weekends. Appreciate that.

She took German in school long ago. If she goes to this school -- classes are in English -- but she can do a 2 week intensive German class that I think would be fabulous.

Normally I do all of my research, learn some of the language, and plan, plan, plan. This was just sprung on us. I need to learn some German phrases! Yikes.

I leave tomorrow, Sunday for my spring fling to europe. Going to Zurich. Staying at Josephine's Guesthouse. Fab location. Women only. I usually like to stay at offbeat places and not going to be there long so thought it might be fun. Booked on booking.com.