Does the same thing happen at the cosmic event horizon (at the edge of the observable universe)?
There is no way to define this. GR doesn't have a general definition of the mass of an extended object. There are definitions of mass such as the Komar mass and the ADM mass, but these only apply to asymptotically flat spacetimes. Cosmological spacetimes aren't asymptotically flat.
When a massive object falls into a black hole it increases its apparent mass.
Here "its" would have to refer to the entire black hole. There is no way to define whether the mass is localized on the horizon. For example, if the ADM mass of a black hole is $m$, that's just telling you a property of the whole thing.
A black hole's event horizon is not really even a place. A place is a timelike surface. The event horizon is a lightlike surface.