Water – Would replacing old mortar for new prevent water from seeping through brickwork

Is it rusted enough that you can pull the old one out? If so, you might be able to find a similar model that could slot right back in the hole, possibly with a little modification. A plastic one might be a good candidate, both for ease of modification and for weather resistance.

If you do manage to do this, I wouldn't try to mortar the new one in, but instead hold it in with Tap-Cons or similar self-tapping concrete screws. A few of these could be driven into the inside brick surface through the mailbox shell, just to keep things from rattling around too much when you open the door.

I would say to wait and see if the water continues to come in after a few days. It is quite possible that the sprinkler water is seeping through above the slab. Are there bathrooms or other water usage utilities above where the leak is? Pipes usually rise vertically from the ground to the bathroom. You could also try to shut off the main water valve for as long as possible to see if the water ingres shows down. Switch off water heaters before you do this. You need to adjust the sprinkler heads so that they are not spraying on the house.