I agree with #3: that drive is probably on the verge of conking and you were lucky to get it to mount again. I wouldn't trust it with any important data, particularly a backup. I'd immediately get a replacement drive.
If this is your ONLY backup drive, I suggest buying TWO drives of (BIG) size (usual recommendation is multiply total disc sizes (not total of what is stored on the discs now) you want to backup times at least 3 or 4) with one being used as a current attached TM drive and the other being a recent TM backup now stored safely offsite. Regularly rotate the two so the one offsite is always a relatively fresh backup. Why do you need TWO? Very common data destroyers like fire-flood-theft will be likely to take out BOTH the master files on the Mac AND the TM drive sitting near it. But one recent TM backup stored elsewhere can make you almost whole again after any such loss.
The key to success with this is that regular rotation. The longer you wait to rotate the 2 drives, the more data will not be recoverable from the offsite drive in a worst-case scenario.
If I want to just have some kind of faith that the Passport drive still has legs, I might reformat it and still not store anything important on it for the next few months to see how it does. Or I might make it a THIRD TM backup drive, considered long-term storage with less frequent "rotation" with the other two... like also backing up to what are otherwise retired drives that still work just for one more backup.