Unfortunately there is no way around this. The password protection is embedded into the PST file itself. If the password is not entered when Outlook attempts to access the file the first time it is opened, then Outlook will not be able to access the file without doing the close/re-open steps. Outlook will likewise not be able to process any rules that require it to access the file. It's a limitation of Outlook and PST files.

There may be a way to accomplish this using a third-party add-in or program, but using Outlook's native features has the limitations you've experienced.

-SW