I was hitting this same problem, both when updating 1Password, and when trying to reinstall it. This is on Windows 11.
The fix for me was to add a Microsoft Defender exclusions for %localappdata%\1Password. Here's the installation log for a failing attempt:
INFO 2024-01-25T19:34:27.762 main(ThreadId(1)) [1P:op-windows-starter\src\installer.rs:621] Setup logs path set to : C:\Users\$USER\AppData\Local\1Password\logs\setup
INFO 2024-01-25T19:34:27.768 main(ThreadId(1)) [1P:op-windows-starter\src\installer.rs:78] executing C:\Users\$USER\Downloads\1PasswordSetup-latest.exe, build 81024044
INFO 2024-01-25T19:34:27.768 main(ThreadId(1)) [1P:op-localization\src\lib.rs:239] system locale detected as 'en-US'
INFO 2024-01-25T19:34:27.768 main(ThreadId(1)) [1P:op-localization\src\lib.rs:265] selected translations for EN_US based on detected locale en-US
INFO 2024-01-25T19:34:28.719 main(ThreadId(1)) [1P:op-windows-starter\src\installer.rs:536] Starting fresh install to 81024044
INFO 2024-01-25T19:34:28.910 main(ThreadId(1)) [1P:op-windows-starter\src\installer.rs:302] Copying files to C:\Users\$USER\AppData\Local\1Password\update
INFO 2024-01-25T19:34:28.910 main(ThreadId(1)) [1P:op-windows-starter\src\transaction.rs:71] copying files to: C:\Users\$USER\AppData\Local\1Password\update\8
INFO 2024-01-25T19:34:30.387 main(ThreadId(1)) [1P:op-windows-starter\src\installer.rs:306] Renaming C:\Users\$USER\AppData\Local\1Password\app to C:\Users\$USER\AppData\Local\1Password\previous
INFO 2024-01-25T19:34:30.387 main(ThreadId(1)) [1P:op-windows-starter\src\transaction.rs:128] renaming C:\Users\$USER\AppData\Local\1Password\app to C:\Users\$USER\AppData\Local\1Password\previous
INFO 2024-01-25T19:34:30.550 main(ThreadId(1)) [1P:op-windows-starter\src\transaction.rs:171] renaming C:\Users\$USER\AppData\Local\1Password\app to C:\Users\$USER\AppData\Local\1Password\previous
INFO 2024-01-25T19:34:30.550 main(ThreadId(1)) [1P:op-windows-starter\src\installer.rs:310] Renaming C:\Users\$USER\AppData\Local\1Password\update to C:\Users\$USER\AppData\Local\1Password\app
INFO 2024-01-25T19:34:30.551 main(ThreadId(1)) [1P:op-windows-starter\src\transaction.rs:171] renaming C:\Users\$USER\AppData\Local\1Password\update to C:\Users\$USER\AppData\Local\1Password\app
ERROR 2024-01-25T19:34:30.552 main(ThreadId(1)) [1P:op-windows-starter\src\transaction.rs:57] failed to rollback rename of C:\Users\$USER\AppData\Local\1Password\app to C:\Users\$USER\AppData\Local\1Password\previous. Error: Os { code: 2, kind: NotFound, message: "The system cannot find the file specified." }
ERROR 2024-01-25T19:34:30.554 main(ThreadId(1)) [1P:op-windows-starter\src\transaction.rs:51] failed to rollback update folder creation: C:\Users\$USER\AppData\Local\1Password\update. Error: Os { code: 32, kind: Uncategorized, message: "The process cannot access the file because it is being used by another process." }
ERROR 2024-01-25T19:34:30.554 main(ThreadId(1)) [1P:op-windows-starter\src\installer.rs:183] failed to move app files. Error: Io(PermissionDenied)
INFO 2024-01-25T19:36:43.276 main(ThreadId(1)) [1P:op-windows-starter\src\installer.rs:374] starting C:\Users\$USER\AppData\Local\1Password\app\8\1Password.exe
ERROR 2024-01-25T19:36:43.276 main(ThreadId(1)) [1P:op-windows-starter\src\installer.rs:215] Failed to start previous app. Error: Os { code: 3, kind: NotFound, message: "The system cannot find the path specified." }
ERROR 2024-01-25T19:36:43.276 main(ThreadId(1)) [1P:foundation\op-windows\src\windows\controls\window_class.rs:248] Windows Error Win32(0x00000584): Class still has open windows.