Framing interior non-bearing wall with 24″ on center for sound proofing

First off, a reality check; not trying to discourage you, just trying to set some expectations here:

• First off, the actual pressures you're dealing with are very small. Absolute pressure is measured in Pascals; 1Pa = .0001psi. Your 80dB sound is inducing pressure changes of approximately .2Pa = .00002psi (that's two hundred-thousandths of a psi).
• Second, the percentage changes you need to make to these small pressures are very large. The Bel scale is logarithmic, based on a "reference" pressure. One Bel - 10dB - represents a tenfold increase in pressure. So, a 90% reduction in the pressure of your 80dB sound only reduces it to 70dB; to get the sound level below 10dB (you or I would call that "silence"; in laboratory conditions the threshold for human hearing is 0dB) would require reducing the pressure differential by 99.99999%.
• This level of isolation is possible but extremely cost-prohibitive in most applications; million-dollar recording/mixing studio spaces are built with lower levels of soundproofing than this. You normally see this level of isolation in medical and sound engineering research labs, where they literally build a room within a room, suspended on vibration-dampening struts, with a near-vacuum maintained between outer and inner walls of the chamber.

The upshot of all of this is that, no, a reduction in sound of 80dB is just not possible on a $1000 budget.

Now, all that said, don't lose heart. First, you seem to have much more ability to make changes to your rented space than the average apartment dweller would have (though I'd double-check your rental agreement; if you're required to return the space to its original condition - or pay damages - when you leave, think carefully about studding in a new wall).

Second, much smaller reductions in SPL can make a very big difference. Soundproofing that reduces an outside noise by 20dB makes the noise sound only a fourth as loud. This is easily doable in your budget (add some sound-insulating material such as mass-loaded vinyl between the existing wall and an extra sheet of drywall or wall covering), add a sound-insulating curtain or three, etc and you'll start hearing results pretty quickly. Two layers of mass-loaded vinyl will reduce most sounds above 125Hz by 18dB just by itself.

It's impossible to say how soundproof an installation will be. We can make a reasonable estimate if we know all the details, including potential flanking paths. A wall of this type may give you an STC 45 or something (wild guess). But even if it were STC 300, you'll still hear a screaming child due to flanking paths. Just how much do you need to hear? You're not going to hear them plotting their first great escape, but if a child is in trouble, you'll hear about it, unless the stereo or TV is up fairly loud.