Before you go to see it, you should ask the seller if it's already plugged and if it's not, ask that it be plugged and running on the coldest setting for two days before you see it.
Visually inspect it for damaged seals, plastics, missing parts like door shelf retainers, cabinet rust, etc. (Frigidaire refrigerators don't rust that much compared to other brands but it can happen).
Have a thermometer (or two!) to check the temperatures in the freezer and food sections. You want the freezer to be at least a few degrees below 0°F at the coldest setting, the refrigerator section could be too cold too, around or below 32°F. These refrigerators defrost their freezers twice a day so unless the defrost cycle took place in the last 2-3 hours, that's what you should see. Turning the cold control towards "off" should make the compressor cycle off much before you reach the "off" position. Turning it back to the coldest setting might not make it start again immediately if the temperatures are still very cold inside, leave the doors open and wait until it restarts.
Look for water dripping or frost at the top of the refrigerator section. If there's a trace of water or frost, it's likely caused by bad (wet) foam insulation under the evaporator. That's a common issue with old Frost-Proof refrigerators and it's usually not a big problem to fix but it can be annoying if not fixed. The easy way to fix that is to stick a heated wire underneath the foam so it dries the insulation before it drips. The other way is to replace the insulation but that's not easy and finding the replacement molded foam might be impossible or almost!
Then, look for noises from the fan motor (there's just one, located in the freezer in this model), the compressor, look for oil leaks at the electrical connections under the compressor. If it's noisy, it could be a tired compressor but it could also be caused by hardened rubber mounts, there are two for the compressor and another one at the front for the superheat coil. The defrost timer could be noisy too. It's located under the refrigerator accessible when you remove the lower aluminum trim plate (to remove, open the refrigerator door, grasp the lower section and pull on it, then just lift it out of it's top anchors).
If you have a multimeter, you can have an extension cord with a ground connection, and plug one probe of the multimeter in the ground connection of the extension cord and the other probe on an unpainted part of the refrigerator (it could be a cabinet screw) to see if there's a current leak (you need to have the multimeter set to 200V AC). Reverse the plug of the refrigerator in the outlet (the 1963 models still have no ground or polarized plugs) and test again to see if there's a current leak. If there's one, it's likely the defrost heater that is shorted to the ground but it could be something else too!
Then you might want to manually advance the defrost timer to see if the defrost cycle works. First, look through the freezer air inlets (under the freezer floor) with a flashlight to see if there's any visible frost on the evaporator. If there is, it should melt during the defrost cycle.
If it still has the original style defrost timer, there's a metallic shaft that's visible from the front of the refrigerator (with the aluminum trim still removed). Turn it clockwise until you hear it "Click" and the compressor should stop. Then the defrost heater should melt the ice on the evaporator and the freezer floor might get warm in the next few minutes. Wait until the compressor restarts by itself at the end of the defrost cycle (that should take 10-18 minutes). Then, after 32 minutes, the defrost timer should reset back to "normal" and defrost again in 12 hours. (you should check that again!).
I just took a few pics of my 1964 Frigidaire at home. The defrost cycle just took place a few minutes before I took the pics, the temperature in the freezer was still below 0°F (-20°C) as there is some food in the freezer that acts as a ballast but if it would be empty, the temperature would have gone up quite a bit.
The second picture shows the evaporator fins that are barely visible but clear of frost under the plastic air inlet grille in the freezer.
The third and 4th pictures show the original type defrost timers in two different fridges. It's at the same location on the 1963 models. The red arrow on the 4th picture shows the advance shaft of the timer.
I have no idea about how much you're going to pay for this fridge but if it's local and priced reasonably, you might want to do some of these tests at home before you put it in service, not at the seller's place!
