Why did her fingers suddenly turn old?

It's Rinnosuke's imagination.

Or perhaps the fingers belong to someone else. Her father maybe, who left it with him for safekeeping.

the old fingers is marisa, she is human and rinnosuke is half-youkai

I gonna guess that the "thing" Marisa cooked has given Rinno visions of both Past and "Future".

cool but i didnt understand what was going on half the time