It's possible to court Nina at night, when time doesn't pass, allowing you to raise her affection as high as you want within the first week. Once you have the fishing rod, you can keep using it for as long as you like, provided you don't talk to the fisherman or leave the area. Normally, there isn't much you can do with a fish without leaving the mountain. However, on Sunday afternoons, Nina can be found at the top left corner of the mountain, and she likes fish. It may take a while in realtime, but you can catch and bring her as many fish as you like, raising her affection as high as you want in a single Sunday.