Yes you are correct, I was thinking about HSRP. VRRP does allow you to use the physical IP address of a particular router as the virtual address that is shared among all devices in the VRRP group.
So to answer your original question:
The simple answer is yes, SW2 will adopt the IP address of SW1. More precisely, it adopts the role of the virtual router. In more detail, what SW2 will do is it will begin responding to ARP requests informing that the IP address 192.168.1.1 now corresponds to a different MAC address, that if itself.
With VRRP, a virtual router must use 00-00-5E-00-01-XX as its Media Access Control (MAC) address. The last byte of the address (XX) is the Virtual Router IDentifier (VRID), which is different for each virtual router in the network. This address is used by only one physical router at a time, and the currently active device that is playing the role of the virtual router will reply with this MAC address when an ARP request is sent for the virtual router’s IP address.
I hope this has been helpful!