The formal Catholic teaching rests on a very interesting point, that once understood ties a lot of things together.  

This point is that God isn't bound by space and time.  

Some Protestants get themselves wrapped in loops trying to explain Scripture and matters such as predestination, free will, etc.  And many Catholics do too.

Jesus's sacrifice occurred in space and time, but its effects aren't.   

The role and idea of priesthood in the OT wasn't wrong, so much as it was imperfect.   Jesus became the perfect priest (and perfect victim).

So Catholic priests participate in the priesthood of Jesus Christ by lending God their all. 

So the priesthood continues.

And the "common priesthood" (you and me) continues as well. We can share in the intimate union with Jesus by offering Him our work, our struggles, our joys, etc.   We "partake" in this act of charity with Him, just as Simon of Cyrene did (in a particular point in space and time).