Salt does work to depress the melting temperature of water, but does not necessarily give you a longer "cold cooler" time. The reason that it works and also the reason ice is a great way to keep stored foods cold is because water is undergoing a phase change, from a solid to a liquid. During the phase change process, it takes more energy to warm up the block. The block will go from 0F to 32 F at a faster rate than it will go from 32F to 33F during the phase change.
Salt water pushes that phase change "energy bonus" to a lower temperature. It is most useful when you have food that you want to keep colder than freezing, and that's also why it is used to make homemade ice cream, as it keeps the melt point at a lower temperature. Also salt water does not have a constant
X degree temperature (like 32 F for normal water is constant) during phase change, it varies, which gets rid of some of its colder temperature bonus for storing the cold for long periods. And unfortunately, the colder the temperature inside the cooler, the more differential temperature there is. A 32 degree cooler on a 70 degree day has a 38 degree differential. A 4 degree cooler inside has a 66 degree differential, and gains heat through the cooler walls at a much faster rate because of that. Which brings me to the next point!
A block of ice is preferred for coolers why? If you have a 10lb. block and 10lbs. of cubes, what's the difference? Surface area! The more surface that is in contact with the air in the cooler, the faster the ice can gain heat by convection between the two materials. Cube ice is much more likely to get your cooler colder, faster, than a big block due to this. But it will also pass through the phase change faster. Often times you will find that the big block will keep your cooler much closer to a refrigerator temperature than a freezer temp, somewhere around 40 degrees. This also slows down the transfer of heat across the cooler's insulation to the outside air, because the differential temperature is lower! If your cooler is too warm but the block is not melting very fast, just make several, smaller blocks of ice and your cooler temp will be colder next time, and your ice will melt a bit faster.
Which block will last longer? Salt or no salt? Well, you have a complex problem, that depends on the size and number of your ice blocks, and the R-value of your cooler as well. If it is exposed to radiant heating from the sun, most coolers do not have a foil backed insulation to protect them from that, but you can make your own! If you like to experiment, you can test your cooler and ice blocks in the back yard with a thermometer and a timer, or just watch the clock. If you are geeky, you can set up a data logger. So much fun!
Here is a little paper I searched Google and found, it discussed the basic differences between specific heat and latent heat in phase change materials, like ice!
Want to make a radiant insulated cooler cozy?
You might try this stuff!Also, are you going to climb the Himalayas and need to keep your beer cold? Well, that should not be a problem in the Himalayas - but for sake of argument, at a higher elevation, the freezing point of water is just a little bit higher, due to the lower air pressure. This is different than most phase change materials because water expands when it freezes which is unusual. But anyway, the higher melting point means your differential temperature might be a bit better, and you'll get a few more minutes of cold beer! If you want to super geek out, you need to know about the specific heat, the latent heat, and the change in volume of a substance during its phase change!
Even more fun! What is the best shape to have the most interior volume but the least surface area to slow down convection heat transfer? It's a sphere! So the next time you are doing your cooler experiments, freeze some water in a latex balloon and see what the difference is!