First, the word "fail" is out of place here. Fail is the opposite of succeed, not the opposite of complete.
Second, in the U.S., we would say "the help of a computer". I know in other places, like England, they talk funny and leave out things for no better reason than they can. Like "I went to hospital" instead of "I went to the hospital". Granted, in the U.S., we say "I went to school." But so what?
Third, without is a preposition. Technically, the rule is that you don't end a sentence with a preposition. However, it's rather more awkward to say "That's the only one of which I can think" than "That's the only one I can think of." So, we often end sentences with a preposition.
Another problem is the carried verb: "I can succeed with the help of a computer, but fail without". "can succeed" would carry to "can fail". Maybe okay, if that's what you mean. If you mean you will fail without a computer, then you can't omit the "will".
"I can succeed with the help of a computer, but will fail without." This just isn't like "only thing I can think of".
"I can succeed with the help of a computer, but will fail without it." This would be okay if "it" unambiguously referred to something. But one might wonder whether "it" referred to "help" or "computer." What is "help of a" doing there anyway?
"I can succeed with a computer, but will fail without one."
If you need to refer to "the job", then "I need to use a computer to complete the job." There is the necessary versus sufficient problem. A computer is necessary. Is it sufficient? If it is, then "All I need to complete the job is the use of a computer."