So its connected as an external hard drive then.... you should be able to just copy the PST file across from the drive acting as an external to this other healthy computer hard drive. However; if data that is damaged on this drive is a part of the composition of this PST file you can run into problems.
Software I have used for data recovery is not free but it works to get most of the data back. No guarantee that this PST file isnt damaged beyond repair. I own a licensed copy of GetDataBack NTFS. This has been pretty successful at getting clients data back, however if the data is located in a damaged portion of the drive, the data could be damaged beyond repair.
https://www.runtime.org/data-recovery-software.htm*Note: The demo will show proof that it can find files, but in order to get this software to allow copying of the data from damaged drive to healthy drive they want you to pay for it.
Charging people for data recovery and using this software with a spare computer, I was able to make my money back and a small profit using this software, but I have never gotten 100% of everyones data back with it usually about 95% of it but the data that is located in physically damaged areas of the drive is trashed. What is impressive is that a hard drive that had the clunk of death, this software was able to gather the data as the heads sweeped the drive during the clunk of death and it was able to reconstruct the data on a healthy hard drive. It took almost a week of this running to get the data back from the drive with the clunk of death.