If the subject is singular, do you change it to "needs not"? For example, would it be "He need not send a letter" or "He needs not send a letter"?
Always "need not." Always.
This is a great question! While the expression "need not" isn't particularly common in modern English, if you want to use it, remember that it's always "need not." Yes, this does somewhat defy the laws of grammar. But since it's an expression, such exceptions can be made.
Although, a more modern way of phrasing your example sentence would be "He doesn't need to send a letter." This sounds better and flows a lot more smoothly, unless you're specifically going for that older writing style.