Seat and business address are two separate entries. The seat is a municipality, Vienna for example, and is set in the articles of association. The business address is the specific address with street and house number. According to the Business Service Portal of the Republic of Austria (USP), this delivery address is recorded in the commercial register as well, separately from the seat.
What matters is deliverability. The Commercial Register Act lists what has to be recorded and names the seat and the business address relevant for service. Proof of ownership of the property is not among them. Renting an address and sharing it with other companies is therefore permitted. At Postservice.at we receive the mail for every company at the address.
Your rental agreement serves as evidence. Within one business day you receive the address holder's signed consent, and your notary files it with the commercial register application. So the lease comes first, and the founding application already names the Vienna address.
A later change runs through the commercial register court. If the new address stays within the same municipality, the seat does not change and a simplified filing without notarial certification is enough. If the municipality changes, the articles of association have to be amended first and notarial certification is added.