It's fine, but a bit formal and old fashioned to my ear as an American. You'll most often hear "I don't need to come."

'I needn't come' is indeed correct. More commonly used in British English than American English. And it is used when a person is asked if they need to be somewhere. It isn't used as a common answer. It is reserved more for instructions and is usually second person; 'you needn't come.' (E.G., someone tells you you need to fill out a form but you needn't come to the office to submit it). I would use 'I needn't come' if I was probably in the above example and relaying what I was told to someone else. 'I need to fill out this form, but they said I needn't come to submit it'. Hope this helps.