Are there are any Segulot, Tefilot, Limudim brought down in the Sefarim for the passing of the Sadik HaDor?
ברוך אתה יקוק אלוקינו מלך העולם דיין האמת.
Furthermore, if the deceased was a scholar to the extent that one could ask him a question in all matters of Torah and he knew right away, then the Shulchan Aruch rules (YD 340:7) that one must rend one's clothing in mourning upon hearing the news, provided it is within 30 days of the funeral. The Rama there only requires clothes rending if one had learnt some of the commentaries composed by the deceased.
Rav Yonasan Eybeschutz writes in a place I can't remember that when a tzaddik passes away, his middos become hefker (ownerless) and anyone can acquire those middos more easily. So one should focus on acquiring the middos in which that tzaddik excelled.