Refrigerator noise

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.

.I assume that you will need to be able to use that passage and the walls as you said are soundproof enough. Then you need something like a door or a hanging curtain. To stop the sound you would need some mass and you can also add some absorption into your room. I suggest you start with the following: 1 - place the refrigerator on a rubber mat - that will decouple it from the floor and cut sound transmission this way; 2 - hang a heavy curtain. you can use mass loaded vinyl http://www.vocalboothtogo.com/acoustic-and-soundproofing-products-vocal-booths/soundproofing-materials/limp-mass-barrier-mass-loaded-vinyl-for-vocal-booth/6-17

or a Mass Loaded Vinyl with Acoustic foam (fire retardant) : http://www.vocalboothtogo.com/acoustic-and-soundproofing-products-vocal-booths/soundproofing-materials/advanced-soundproofing-and-sound-absorption-panels/6-18

3 - when you hang the curtains you can hang them from BOTH sides of the passage wall this will have two-side effect - it will double the sound blocking mass and it will create layered barrier with layers of different density. 4 - make sure that your curtain overlaps the opening and creates a tight (as much as you can) seal around the edges. if you can see a light through your cover - the sound will leak through it.

5 - and lastly I am not sure looking at your drawings if the refrigerator sits just in one corner of the room, but is you can you can enclose the refrigerator itself (just on the side facing your room with sound absorbing or sound blocking materials. Do not enclose the whole refrigerator – in needs air exchange to function But if you put another barrier in the direction of your room – it will help.

Good luck. Jeff.