I'll definitely flee (along with some people I could trust) to my family's mountain cabin and hunker down. The place is self-sustaining with solar panels and water from a nearby stream, and there's a patch of land sizable enough for me (and others) to make a farm.
Then I'll place traps (pitfalls, bear traps, deadfalls, caltrops, even explosives since I know how to make one) all over the forest and on the road towards the cabin to ward off intruders.
We'll only go down the mountain either to scavenge for materials, tools and equipment, to reconnoiter for enemy movements and look for more survivors, or to interact, trade and communicate with other survivor bases.
Considering that this is an invasion, even if the Mamono armies start at small towns, they'll still get noticed by the governments, since information gets around very quickly and a town suddenly going dark never bodes well for anyone. Once their intentions to invade are verified, all nations will go to war status and attempt to suppress them.
But combine the Mamonos' physical superiority, access to magic, and the fact that they have modern weapons as well, the situation is heavily tipped to their favor and inflict heavy losses on our side. Theoretically, humanity will get desperate enough to start lifting any international law regarding weapon restrictions and regulations, including WMDs.
It will devolve into a MAD situation where for every city lost to the Mamono, humanity will pay them back with nerve gases, nukes and engineered plagues.
Asset/Area denial tactics will be used as well. Landmines and caltrops will now be a common sight, and the remaining human cities will have megaton-yield nukes sitting right on them, all primed and ready to detonate in the event that their defenses get overrun.
But it's just theoretical.