hmm.. its just that the oil cooler leaked for some reason.. its not exactly that you did something wrong.. they just tend to go for no reason (most of the times)..
So your car has a built-in radiator oil cooler? To confirm, see the pipes going into the radiator. There are two big hoses for water, two small hoses would be for the trans oil.
What I'd do is, leave the tranny cooler in the radiator like that, seal the outlets with compatible screws and get the radiator flushed. I'd buy an external trans oil cooler, put it infront or in the back of the radiator and possibly behind a fan pushing air in. I guess your car has two fans for cooling. Again I don't know about city's layout, but I'd get the trans oil filter changed / cleaned, clean the oil pan, flush as much as I can out of, fix everything back, start putting in CHEAP ATF untill fresh ATF comes out of the drain, drive and check the car.. if everything seems to be fine, buy a good ATF then flush / change.
Now again, I'm not the most practical guy in the world. If you don't want to get into buying / importing an external oil cooler, you just get a new OEM one for your car, goto a radiator wala, get the radiator flushed / new oil cooler put in and be done with it. rest of the process remains the same.. until or unless you want to get the tranny out. OR if you have to get the tranny out because its effed now. Let's hope it will be good once again when you drain water out of it and put in fresh ATF.
Good luck. But dude, first confirm that your radiator has a built-in tranny oil cooler, and other people who have or know about city vario, comment on this if possible.