How to stick acoustic foams on the walls

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.

If you're only going with one or the other, my personal opinion would be to go with accoustical drywall and seal the perimiter with accoustical caulk. Insulation goes between the studs, so you could still have vibrations traveling through the studs themselves and the top and bottom of the wall's frame. If you use sound-deadening drywall like QuietRock or use a double layer of regular drywall with Green Glue in the middle, you can avoid that flanking much easier since you're covering the entire area of the wall's frame. Of course, worst-case sound situations could still allow sound to flank through the floor or ceiling, under or over the frame, and through the floor or ceiling of the neighboring room, but I'd think that would be an extreme case.