Swiss Saver Day Pass and a border crossing

When you put your toe across the Swiss border on departure the Swiss pass becomes invalid.

So before you jump on the train just go to the Swiss ticket counter and tell them where you want to go, show them your day pass and they’ll tell you what your options are.

Swiss Passes and special tickets are valid up to the last Swiss station, i.e. the last station run by a Swiss railway company, which may be before or after the border.
Where are you actually travelling from and to (which stations)?

"specifically, I will be crossing from Switzerland into Austria at Feldkirch." - on that route, the last Swiss station is Buchs(SG), where the Swiss crew gets off and the Austrian crew gets on. There are then other stations in Liechtenstein, which the long distance trains don't stop at, before getting to Feldkirch.

"and then I just get off and purchase a point to point to my destination?" - that would be a waste of time, you would then miss the train and have to get the next one.

I would try pricing this trip, as one purchase from your start to your destination station, on both SBB (Swiss railways) and ÖBB (Austrian Railways) websites. These should apply both SBB and ÖBB advance purchase discounts and give you a price for the complete trip.

"What do I do at the border?" - nothing, just stay seated on the train, provided you have bought your tickets ahead of time.

At Chris F, specifically traveling from Zurich Flughaven to Bludenz, or at least contemplating it. So, I would need to have, in addition to the Saver Pass, a ticket from the last station on the Swiss side to Bludenz I take it? I don't know if I will be able to determine what that last stop is via the excellent SBB app. I guess I may have to discuss with the attendant.

Michael, I went to the SBB website: https://www.sbb.ch/en
You could equally do this on the App, but my phone is in the other room!
Entered Zürich Flughafen to Bludenz, and a random date and time. Click on one of the options to expand it and show the trains. To pick one example:

Zürich Flughafen dep 10:18, Zürich HB arr 10:27
Zürich HB dep 10:37, Sargans arr 11:32
Sargans dep 11:37, Bludenz arr 12:28

Now, click on "Show on map" and you can see the route, and where it crosses the border.
There are two possible routes, some cross the border at Buchs(SG),some at St Margrethen.
The ticket office at Zürich Flughafen station can sell you the ticket.

However, this confuses me:

So, I would need to have, in addition to the Saver Pass, a ticket from the last station on the Swiss side to Bludenz I take it?

If you have just flown in (why else would you be starting from Zürich airport?) and are immediately leaving the country,why would you buy a Swiss Pass?
If this is the only trip you are doing that day, just buy a ticket from Zürich Flughafen to Bludenz. One ticket.
You can buy this when you land. If you buy on the day, it will cost about CHF 52 (the SBB website told me that).

OP said they bought a Saver Day Pass, not a Swiss Pass

Oh, sorry, I read the 2nd post where the OP just says "Saver Pass"
Still my point applies. Why buy 2 tickets, Swiss + Austrian. Just buy 1 ticket From Zürich Flughafen to Bludenz, at the airport station when you land. Simpler.

"I guess I may have to discuss with the attendant." I hope the OP doesn't mean buy a ticket on the train - not possible.

Well I guess if he decides to talk to the attendant on the train he'll not quickly forget the answer :-)

"Still my point applies. Why buy 2 tickets, Swiss + Austrian. Just buy 1 ticket From Zürich Flughafen to Bludenz, at the airport station when you land. Simpler."

I bought the saver day pass several weeks ago because I was going to Zermatt, and the Saver Day Pass was significantly cheaper than a point to point ticket. Then the group I was meeting in Zermatt cancelled due to coronavirus concerns. Rather than a long trip to Zermatt followed by a longer trip to Klosters to meet some other friends, I'm going to go somewhere closer to Zurich Fh and Klosters. But I have the (as I understand) non-refundable Saver Day Pass, so may as well get a little use for it.

Travel happens.

Michael, it looks like you can buy the ticket for the Austrian portion on the OEB website ahead of time, https://www.oebb.at/en/ Feldkirch - Bludenz , for @ €5.20 - now that you know the last Swiss station. It wouldn’t sell it from the last Swiss station, though.

If you just buy a ticket Feldkirch - Bludenz, you are still missing the section Buchs SG - Feldkirch.

I can get both ÖBB (Austrian Railways: https://www.oebb.at/en/ ) and SBB (Swiss Railways: https://www.sbb.ch/en ) websites to sell me a ticket Buchs SG - Bludenz provided it was a train. Some of the connections listed are buses, and it wouldn't sell me a ticket for those.
I would recommend you buy your ticket Buchs SG - Bludenz when you land at Zürich airport. You cannot predict exactly when you will land and get out of the airport, so you cannot predict which train you will get until you get there.