I'm not familiar with the markets in Cork, because i'm from Limerick, however I do know about the best stores to shop at. Aldi and Lidl are known for their good fruit and a cheaper price, although I don't know how accurate this is because I don't go there...there isn't any close, where I live in Limerick. I find Tesco and Dunnes to be good, especially later in the evening. They usually have a clearance for the fruit/veg..sometimes you can get really good deals, such as lettuce for 10c or a pack of mangos for 1euro. Make sure you check them carefully though, because some of the fruit/veg is fermented.
I found dates to be difficult to find, and when I found them (at Tesco) they were a bit overpriced imo. Try and bring some in your suitcase to tie you over. That's what I do and I never get caught coming through the border. Even if I did, the worst thing that could happen is they'd dump them. Other than that, you'll be fine buying bananas and starches.
Ireland has really upped there game over the past couple of years when it comes to catering to veganism so I wouldn't worry.
Hope this helps, have fun in Cork and enjoy those spuds!! :)