Over friendly neighbour...

I was asked in an interview for a Japanese exchange programme what I would do if when I moved to Japan a local neighbour repeatedly called on me to give me food/meals and how I would deal with it. I didn't get through but want to apply again for the programme; would would be the best thing to do in this situation?

I have read a little on Japanese gift etiquette, but any suggestions on this would be very much appreciated.

It might depend on the situation, but a friendly way to respond is probably to accept them with a word of "Oh, you don't have to do that. (sonna ni ki o tsukawanaide kudasai)" And then, try to give something in return on another occasion. This "something" can be small gifts from your home country, sharing fresh food you can't eat all by yourself, or non-visable service such as offering help around the house.

Is a friendly way to respond an ideal way to respond, then? That's another story.

I hope that your answer to that question wasn't something that decided whether you were selected or not...especially if you don't know J culture and had read only about text-book J customs.
I think that you should politely accept the gift(s) with an "itadakimasu" ('I accept') and afterwards, catch the gift-giver one day and say "senjitsu, gochisousama deshita" ('Thanks for the food the other day, it was great'). Even if you don't/can't eat what is given, or if it was terrible tasting, you should accept/thank them.
But there are many Japanese who flat out refuse gifts of food and it's not a big deal.