The situation in Roman Judea is complicated by widespread bi and trilingualism, and diglossia.
As a hellenised area, Greek was the lingua franca and cultural elites would normally function in Greek. This would include the administration.
As an upper class Roman, Pilate was probably bilingual in Latin and Greek, and would have conducted most of his governance in Greek.
Latin was more prevalent in the military, but given the setting, Greek would indeed be spoken, especially if talking to a local.
The Jewish population was, to varying degrees obviously, functionally fluent in Koine Greek.
Despite his rural origin, it would be extremely unlikely that Jesus knew no Greek.
Did Jesus understand everything at his trial, etc? It's difficult to be certain , but everything would suggest that it would have occured in Greek and he would have been able to follow it.
I'll add some sources on the linguistic-scape of Roman Judea once I'm in at my office this morning.
Edit:
Some further reading:
Mark Janse, 'Bilingualism, Diglossia and Literacy in First-Century Jewish Palestine' - a good starting point with a good bibliography as well.
Stanley Porter, Diglossia and other topics in New Testament linguistics, ed. Sheffield, 2000 - an edited volume that covers some of this material.
Penelope Fewster, 'Bilingualism in Roman Egypt', in Bilingualism in Ancient Society (2002) makes a good parallel case, and interacts with the reason the Roman East was so Greek-dominant - the legacy of Alexander.
On bilingualism among Romans (Latin/Greek), see:
Alex Mullen, Bilingualism and Multilingualism in the Roman World
For the classic case that Jesus spoke Greek, apart from being a native Aramaic speaker, see Josephy A. Fitzmyer, 'The Languages of Palestine in the First Centruy AD'; as well as Fitzmyer, 'Did Jesus Speak Greek?' Biblical Archaeology Review 18/5 Sep/Oct 1992: 58.
Jesus speaking of Aramaic is just assumption based on what a person of his backround would speak it is not actually stated anywhere in the Bible or elsewhere. It is likely the case but I would not call it a fact. According to Bible he lived in Eqypt as well for a time so might have learned languages there (or being thaught if his family learned them). What I mean is that we do not really know anything and it is just speculation what he could speak.