bit of a over statement as long as you seed the tank with a piece of lr which you will want to do because that will have pods,snails,worms, stars ect ect in it already your dead rock should be cycled with in a couple weeks.
this misconception of months to cycle a tank is wrong nither of my tanks took "months" and that was using all dead rock then adding one piece of live rock eg my 240ltr tank i used around 20kg dead rock and 1kg live was cycled in under a month
as for going out and buying live rock you can at most pet shops that sell marine stock where are you located?