Can I travel to Austria on a new visa-free passport if I lost the passport that contained a long-term Austrian visa?

Obviously too late for this poster but here are some details:

Getting a passport

Assuming you're American, make an appointment with the local passport agency and make your travel plans clear - explain that you need your passport ASAP so you can sort out the visa in time. Their site will make clear what supporting documents you need.

Getting a visa

Normally, your exchange program would help you resolve this.

Austria differentiates between people intending to stay less than six month, and those staying longer. For stays of less than six months you are required you get a visa, as was done in this case, this must be done from outside of Austria except when the case is a defined exception.

Since you can't reach the embassy, I'd try calling the hotline at +43 1 531 26 3557 from 8a-12p CET to get more information on if you can be recognized as an exception or what the best process would be.

Alternatively, if you're staying longer

If you are actually planning to stay more than six months, Americans under normal circumstances don't need a visa to enter Austria, but you do need legal grounds to stay there for an extended time: a residence permit. I'm American and was directly enrolled in university in Austria. You can show up under the usual visa-free process any tourist would and then apply for a residence permit. You need to apply for the permit before said visa-free tourist time allowance is up. If you apply for the permit before the end of your visa-free time but the processing isn't complete by the time that 90 days is up, you are fine to stay in Austria but you're not allowed to travel to other Schengen countries until you receive your residence card. (Though there are no borders so enforcement is minimal, I wouldn't risk pushing this.) Here's the link on the government's site:

Folgende weitere Personengruppen können ihren Erstantrag im Inland stellen, wobei auch hier die Antragstellung nicht zur Überschreitung der Dauer des erlaubten visumfreien Aufenthalts berechtigt:

Additionally, the following groups of people may make their initial application in-country, though this does not allow you to exceed the duration of the permitted visa-free stay:

Drittstaatsangehörige, die gemäß § 1 Abs 2 lit i oder j AuslBG oder § 1 Z 5, 7 oder 9 AuslBVO vom Anwendungsbereich des AuslBG ausgenommen sind oder die unter § 1 Z 4 Personengruppenverordnung 2014 – PersGV 2014, BGBl. II Nr. 340/2013, fallen [...] oder eine Aufenthaltsbewilligung "Studierende" beantragen, nach rechtmäßiger Einreise und während ihres rechtmäßigen Aufenthalts

Third country nationals who are excepted from the scope of application of the AuslBG by § 1 Abs 2 lit i or j AuslBG or § 1 Z 5, 7 or 9 AuslBVO or who fall under § 1 Z 4 Personengruppenverordnung 2014 – PersGV 2014, BGBl. II Nr. 340/2013, and who are applying for [...] a "Student" residence permint following legal entry and stay.

Have fun in bureaucrat paradise. ????