In British English it's a 'stick' - it's a 'walking stick'. You could say it is tipped with rubber. The thing at the end is usually called a ferrule, but as that word is derived from the Latin for 'iron' I'm not sure whether you call a rubber one a 'ferrule'!
PS The rubber tip is at the end, but if you're reassembling it you put the tip on the stick; so you can use either preposition, depending on context.
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