The city of Cork is compact and can be explored on foot. The good news is that you definitely won't need a car to explore Cork. You won't even have to rely on public transportation. The city of Cork is compact and very walkable, and it even has the second largest natural port in the world.
There's no doubt that you'll need a car to explore the county, as public transportation, especially in the countryside, isn't incredible, but while you're in the city, it's best to go on foot. There are tons of cafés, great shops and, of course, a pub on almost every corner. Cork also has an exciting music scene. Although you can visit it all year round, I would do it between February and November.
In fact, I would skip the summer months of June to August and try to travel from late spring to early fall. The weather is good in spring and fall and you'll have to deal with fewer tourists. Cork City Council, in collaboration with the Department of Transport, Tourism %26 Sport, has developed a series of four heritage-related hiking trails in the city center, called Cork Walks. Cork is in a fantastic location for day trips, even if you're travelling without a rental car.
Cork is quite accessible on foot and I ended up staying close to University College in Cork with views of the River Lee.