Yes, he has lost all of his dorsal and a good portion of his tail..
Not saying this to be mean, but it can take months+ for rot to reach the body. Once it does it's iffy if he can be saved.. it's bacteria attacking him from the outside working it's way in.
In a tank that size, with no filter 1 50% and 1 100% water change per week + water conditioner is needed to keep the water properly clean. If anything at least 2 50% per week with siphoning of the waste.. fin rot occurs from a damaged fin and water not being cleaned enough.. bacteria sets in and grows, eating away at the fins and in time, the body.
For now what I would suggest if you want to try to save him is:
2 teaspoons of Aquarium Salt per gallon, make sure to dissolve it before placing him in it - hot water (a drop of conditioner) with continuously stirring (that part sucks) works best to dissolve it.
Daily 100% water changes with new salt/conditioner added in each time. Treatment should last no longer than 10 days.
BUT, this is really advanced, so the salt may not work.. in that case I would look for Maracyn Plus - it can be pricey, but it may work.. others may know of stronger medication, I tend to stick with the more "natural" way to heal.
It's not a for sure thing, his immune system is weakened by the cold water.. so definitely get him a heater soon and treat him, with his age added in.. it's a toss up whether or not he will make it. Medications can be stressful on their delicate organs - even the aquarium salt can shut down their kidneys if left in it too long.
Poor guy.. I would definitely look up the recommended water changes link
for any future fish, it does make a difference. The person who posted it has over 40 years breeding/raising these fish.
If you have any more questions, concerns.. please feel free to ask.