What tabletop, role-playing game rules would be the best fit for adventuring in the Ponies After People universe? I’d love to start a Ponies After People RPG campaign and need some guidance from those who have already run PAP campaigns.

Ponyfinder probably has some rules you could use.

Dead Trees: Fallout Equestria tabletop RPG would also probably help as it has info on all races and is set in a post apocalyptic setting with modernish weapons and so on.

I have run two PaP tabletop games so far, and I used the Savage Worlds rules with the premade pony module: https://www.deviantart.com/giftkrieg23/art/The-Savage-World-of-My-Little-Pony-4th-Edition-347963248

But it really comes down to the level of crunch and complexity you want from a rules system. I wanted something that was easy for my players to pick up, but that means sacraficing mechanical complexity, which some groups might prefer.

The last time I played an MLP game we used FATE for the rule system which is generally somewhat "rules light" but it seems to work well for P&P games hosted online. We played Five Score for a setting instead of PaP but vaguely similar premise of people becoming ponies, just not in a post-apocalypse.

Genesys

I don't know what TRPGs you might have run/played, but if you'd like to see what Pony games have been done there is a list up at https://forums.equestriadaily.com/index.php?/topic/1875-list-of-pony-related-tabletop-rpgs/