Short answer, no.
The method you are suggesting here is to use Joule-Thompson expansion to cool the air as it escapes and hope that it liquefies. This is a valid way to make liquid nitrogen and is used in both lab-scale and industrial air liquefaction. Unfortunately, you will be many orders of magnitude off from where you need to be. Plans I've seen for liquefying air on a lab scale require starting pressures of around 3000 psi. That air is sent through a long copper tube where it is rapidly expanded through a nozzle into an insulated vessel to atmospheric pressure (14.7 psi). Even then it does not liquefy. The cooled air is vented through another tube concentric with the first tube, where it chills the pressurized gas entering the vessel. My recollection is that if you start with a full-sized cylinder of nitrogen, after about 1/2 the tank is gone, you will start to condense liquid nitrogen and when the tank is empty, you'll be left with about a half-liter of liquid nitrogen. Those numbers are probably off, but the order of magnitude is correct. (Note: if you try this based on these "instructions", you will probably get yourself killed given the pressures involved).
Nitrogen liquefies at 77 K, or -196 C. Dry ice is only −78.5 C, and my freezer at home sits at about -5 C. I think a PET Coke bottle gets pressurized to 80 psi and a soda keg is rated to 120 psi. As you can see, the temperatures, pressures, and volumes you're likely to be able to achieve at home are off by one or more orders of magnitude.