Walls – How to fill a gap between wall and ceiling

Step 1: Prevent air transmission between the rooms. Sweeps for the door or a heavier door; look at the heating registers and windows; things like that. You're saying the sound is muffled, which implies the high frequencies aren't making it in, so I don't think this is the major problem.

Step 2: Damp the transmitted vibration. Put big heavy bookcases on the wall between the two rooms (which side doesn't matter so much). Blow heavy insulation into the wall between. Hang a heavy quilt on the wall.

Step 3: Break up the sound in the source room. Use natural diffusers (heavy sofas with heavy foam pillows work great, or the aforementioned bookcase with loads of stuff on it), or buy foam diffusers to stick in the corners/edges of the room.

Step 4: Give up and wear sound-cancellation headphones...or install a white-noise generator in your office (either a fan-type one, or a water feature if you like more 'natural' sounds).

See also How Do You Reduce Floor Sound Transmission for more details.

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.