Is it possible to quick freeze half of an unopened bottle of water or soda by putting half of the bottle in a subfreezing solution liquid nitrogen or other solution without freezing the whole bottle?
I think freezing only half of the bottle would be very difficult. The reason is that water conducts heat very well and better than plastic (I'm assuming the bottle is made of plastic). Moreover, thermal conductivity is not the only heat transport mechanism in water; advection (here: movement of cold water to warmer regions and vice versa) contributes significantly to heat transport. The consequence of water being a good heat conductor compared to the plastic bottle is that all the water inside the bottle will have essentially the same temperature at all times, even though you cool only part of the bottle.
If you find the above a bit technical, think of it like this: The "cold" creeps slowly through the plastic of the bottle, but as soon as it enters the water, it moves much faster, and the "cold" spreads out evenly inside the bottle.
Finally, a word of caution: The water will expand as it freezes, and as you say you want to use an unopened bottle, the bottle may break. If you want to try this, be prepared for a broken bottle (and do not use glass bottles!). The experiment would be safe if you could open the bottle just a little.