The mailbox was on the ground when I came home from work. One sheet was still inside the post and one had snapped off the tip of it but the rest was attached to the screw that was in the bracket attached to the mailbox.

I think I need to buy a hammer drill and I saw good reviews on amazon for one plus some bosch masonry bits.

I guess what I'm not sure about is if it's worth cementing in (some strong adhesive, even gorilla glue) the screws after putting it in again? What types do I buy to screw into the new holes?

In a lot of cases, the advise would be to use a stronger anchoring system, but if you look at it another way, it's better that the box wasn't so secure that it damaged or cracked the post. The plastic sleeve is enough to keep it in place, but obviously not so secure that it took the post with it. Kind of a break away fastening system without out that being it's full intention. I would replace the plastc sleeves and re-mount as originally installed.

I would try pretty hard to get the screws and plastic out before I started hammering and drilling new holes. What I am thinking about right now are:

screw extractors

heat the screw with a solder iron (or very small torch like the type sold for soldering) and pull it out if you can get pliers on it.

solvent to dissolve the plastic. Note that some plastics. like polypropylene will not readily dissolve. I don't know what those things are made of

Overkill to buy a hammerdrill, etc. for this. Better and safer to get the screws out, then reuse the holes. I would drill the plastic anchor with a small drill, then partially screw in a small screw, then grab the screw with a pair of pliers and work it out.

Clean out the old holes and use them again.

A broken wood screw removal cutter (a hollow tube that bores a core) may be needed to cut the plastic anchors and broken off screw and remove them.
The remaining shell off the plastic anchor is then easy to pull out.

@sierraeast: great point about us being lucky. a new box or anchor is far less expensive than a new granite post.
@ionized: I agree. i'm going to try and get the plastic sleeve out. I like the idea above about using a larger screw to grab into the plastic. One of them is snapped in there, so pliers or a drill bit are probably my only hope on that one. I don't have a soldering iron or any plastic dissolving solvent (not even sure where to get that or what brand).
@homebound: I agree. If I can avoid using the hammerdrill, I'm all for it. knowing me, I'd make the post not so pretty. But I'm going to keep the hammerdrill. I need it for screwing into cement wall for shelving in the basement.
@kentvogel: I don't think you read my post and I'm thinking this is just a plug for your blog.
@brickeyee: can you point to a picture of the tool you're talking about or where to buy one?

How do I knwo what size sleeve/anchor to use when replacing it? Measure the diameter of the hole and find something slightly smaller? never used one before.

Link below

Here is a link that might be useful: Broken screw extractor

Let me point out that the mailbox, most likely did not fall down/out of its own accord. Yes, for many years, many youthful "hoodlums" have had sport with destroying mailboxes while out cruising after parties or home games, etc. S__t happens. If you're thinking of repairing your mailbox with something harder to destroy, you'll find it is also harder to fix. And, surely, that you will have to do. So, find a solution that makes it easier to fix and NOT something that is harder to destroy. Remember the "Cold War?" It only becomes a game after a while. Just remember your youth.

@ RRM1: Thanks, but actually there's no damage at all to the mailbox. No scratches either. Simply put, I think the screws and/or sleeves came loose in the post, is all. It's possible they bumped it, but I saw it loose one day and the next it was on the ground.

I Really think you're over thinking this. If it's just plastic that's left in the hole, I suggest getting some long (3in?) drywall screws and just screwing them into the plastic in the hole until you bite, then grab the end with pliers or vise grips and pull back. If the plastic breaks into bits, try again. Keep doing this and blow the loose pieces out the hole with compressed air (a can of computer dust air should work fine) or even a rubber bulb type baby aspirator).

Once it's clean, install new sleeve fasteners.