You can't get anywhere near that close with a single Jupiter sling. It will most probably take decades or even centuries.
Actually, you can. I'll have an example for you and the OP later today.
With that said ... Gentlemen, what is your point? Seriously.
This thread started because someone saw something he thought was pretty cool on a fictional television show that features faster-than-light travel, aliens who all breathe Earth-style air and speak perfect English, and numerous other absurdities. In this show, a spacecraft got very near a star, did a big propulsive maneuver, and got a big impact. The show, which was notoriously sloppy with anything approaching real science, incorrectly called the maneuver a "slingshot." The OP took the show at its word, used the same terminology, and asked if it's possible to "slingshot around the sun to gain speed."
The way I see it, there are two ways to answer the OP's question.
1) Ignore the propulsive maneuver in the fictional example and say "no, you can't do a slingshot around the Sun, because in this frame of reference it's not moving"; or,
2) Ignore (or better still, briefly mention) the mistake in terminology, then say "they called it the wrong thing, but if you do like they did and do a big burn near the Sun you will get a big impact."
Now, let me make two things very clear. First ...
all my remarks in this thread, including the promised example, are directed toward the OP and his original question.
And second ... there is absolutely no reason to assume that the OP is anything other than an eight-year-old who is just discovering science, Orbiter, etc. Bearing that in mind, and that the rest of you have already made answer #1 very clear, I'm giving him answer #2. And you have no way of knowing this, but I do have some small experience as a teacher, and based on that experience believe #2 to be the right call. Period.
Now, you all seem to be implying that answers grounded in perspective #2 are somehow wrong, inappropriate, unfair, unrealistic, etc. Well, the OP didn't ask, "how would you even get near the Sun in the first place," "what's the Oberth effect," etc.
So I repeat, what is your point ... other than puerile attempts at the tired old Internet pasttime of "one-up the newbie"?