Get a Smokers Tobacco Pipe Cleaner, It is a flexible piece of wire covered with a fabric. Possibly tie two or three together to give you the length required. Stick it down the drain hole at back of fridge to clear the blockage. I think the pipe is just an overflow that goes nowhere and should be safe enogh . If in doubt turn off the elec power before pushing the wire down the hole. These are also useful for clearing the drain holes at bottom of aluminium windows.
I use cotton buds to clear out the drain hole regularly. If you unblock the hole and the water continues to build, check that the temperature is right in your fridge as if its not cold enough it will create excess water.