You're not going to be able to successfully open a version 9 project in version 12, Tonapah. The program has undergone so much re-programming since version 9 that not only is it not likely to work, but you're very likely going to end up corrupting your version 9 project file so that you won't be able to open it in version 9 either!
So if you do try, be sure to make a copy of your original version 9 project file. So that if and when the project bugs up in Premiere Elements 12, you'll still be able to go back to version 9 and finish the project.
Meantime, the program is also asking for help locating at least one of your original video files. You'll need to follow the prompts to show the program where that and whatever other video or audio files you used in your project are located.
Best advice though: Finish your Premiere Elements project in the version you started it. Very few Premiere Elements project files are backward or forward compatible.
Best of both worlds suggestion perhaps....
Archive your Premiere Elements 9.0/9.0.1 project, using the project's File Menu/Project Archiver/Copy Option. That will save the project file (project.prel) + the source media that went into that project.
Take the saved Copied Folder and transfer it to a USB Flash Drive. Use the USB Flash Drive to transfer/copy its Copied Folder to the second computer.
When you want to open the version 9.0/9.0.1 project in Premiere Elements 12, right click the 9.0/9.0.1 project.prel file, select Open With, Premiere Elements 12.
Once you edit this file from the Copied Folder in Premiere Elements 12, you cannot edit it some more in Premiere Elements 9.0/9.0.1. But you do have the back up for the original project 9.0/9.0.1 project.prel.
It is worth a try.