"He is not such man as would leave his work half done" is the correct phrasing. This is very formal, however. In fact, I can't think of a situation in modern English where it would be appropriate to use this kind of language - perhaps a judge, summing up a case in a formal court of law? In everyday modern English, we'd say something like this: "He is not the kind of man who ..." or "He is not the kind of man that..". NB Both 'who' and 'that' would be wrong in your original sentence.

He is not the type of man, that would leave his work half done.

That

It's actually "who'". He is not a man who would leave his work half undone. Since you're referring to a person, that person is not an object so you should be using who.