All the references come from the same place, and are from about the 1930's.
"É pique, é pique..." is a reference to an university student named Ubirajara Martins, who always had with himself a little scissor to trim his beard and mustache. Because of the sound of the trimming, people greeted him as "pic-pic".
"É hora, é hora..." is a reference to some students who always went out drinking, but had to wait half an hour for the beer to get cold, on some ice bars. When the time came, they cheered "é meia hora, é hora", which translates as "it's half an hour, it's hour".
"Rá-tim-bum" is a reference to an indian guru (or something like this), named Timbum, who cativated the students with the sonority of his name.
Source and more details: «USP 70 — O Brasil que as arcadas vislumbraram», Pesquisa FAPESP 102, Agosto de 2004, p. 57.